#244544

order	512
nilpotency class	3
isoclinism	isoclinic class of #244523
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	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

975 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 975 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,7)(4,8)(5,9)(6,10)(11,17)(12,18)(13,19)(14,27)(15,22)(16,23)(20,25)(21,28)(24,31)(26,30)(29,32)(33,34)(35,44)(36,49)(37,40)(38,43)(39,47)(41,50)(42,53)(45,56)(46,54)(48,55)(51,58)(52,57)(59,60)(61,63)(62,64);(1,3,6,13)(2,7,10,19)(4,11,15,25)(5,12,16,26)(8,20,22,17)(9,18,23,30)(14,24,28,32)(21,31,27,29)(33,35,38,48)(34,39,43,56)(36,46,52,62)(37,45,53,47)(40,44,42,55)(41,54,58,64)(49,59,57,63)(50,61,51,60);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,24)(13,25)(16,28)(18,29)(19,20)(23,27)(26,32)(30,31)(34,40)(35,45)(37,53)(41,49)(42,43)(44,55)(46,60)(47,48)(50,51)(57,58)(59,63)(61,62);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,23)(11,24)(13,26)(15,28)(17,29)(19,30)(20,31)(22,27)(25,32)(33,36,38,52)(34,41,43,58)(35,46,48,62)(37,51,53,50)(39,54,56,64)(40,49,42,57)(44,59,55,63)(45,60,47,61)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,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, 151740);