#331072

order	512
nilpotency class	3
isoclinism	isoclinic class of #331055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

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

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, 238268);