#331936

order	512
nilpotency class	3
isoclinism	isoclinic class of #331927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19509	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

520 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239132);