#249878

order	512
nilpotency class	3
isoclinism	isoclinic class of #249839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61876	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

924 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13)(4,19)(5,23)(6,24)(7,11)(8,12)(9,16)(10,22)(14,31)(15,26)(17,28)(18,27)(20,30)(21,29)(25,32)(33,34,39,44)(35,40,47,57)(36,48,50,60)(37,42,52,56)(38,53,46,43)(41,54,59,51)(45,58,61,63)(49,64,62,55);(1,3)(2,9)(4,20)(5,15)(6,17)(7,16)(8,18)(10,26)(11,27)(12,28)(13,24)(14,21)(19,25)(22,32)(23,31)(29,30)(33,35,39,47)(34,40,44,57)(36,45,50,61)(37,38,52,46)(41,55,59,64)(42,53,56,43)(48,63,60,58)(49,54,62,51);(1,4,8,5)(2,10,12,29)(3,14,18,32)(6,21,7,22)(9,25,28,31)(11,23,24,19)(13,30,27,26)(15,16,20,17)(33,36,39,50)(34,41,44,59)(35,45,47,61)(37,49,52,62)(38,54,46,51)(40,55,57,64)(42,58,56,63)(43,60,53,48);(1,5,8,4)(2,10,12,29)(3,15,18,20)(6,21,7,22)(9,26,28,30)(11,19,24,23)(13,31,27,25)(14,16,32,17)(33,37,39,52)(34,42,44,56)(35,46,47,38)(36,49,50,62)(40,43,57,53)(41,58,59,63)(45,51,61,54)(48,55,60,64)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157074);