#249873

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 960 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) = 128

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

(1,2)(3,13)(4,19)(5,23)(6,11)(7,12)(8,16)(9,21)(10,20)(14,30)(15,25)(17,27)(18,26)(22,29)(24,31)(28,32)(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)(2,8)(4,15)(5,17)(6,16)(7,18)(9,28)(10,25)(11,26)(12,27)(13,23)(14,20)(19,24)(21,31)(22,32)(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)(2,9)(3,14)(5,20)(6,21)(7,22)(8,24)(10,12)(11,29)(13,25)(15,16)(17,32)(18,31)(19,23)(26,28)(27,30)(33,36)(34,41)(35,45)(37,49)(38,54)(39,50)(40,55)(42,58)(43,60)(44,59)(46,51)(47,61)(48,53)(52,62)(56,63)(57,64);(1,4)(2,10)(3,15)(5,21)(6,20)(7,22)(8,25)(9,12)(11,29)(13,24)(14,16)(17,31)(18,32)(19,23)(26,30)(27,28)(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,d^2=i,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, 157069);