#331935

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	#19510	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)(4,8)(5,9)(6,11)(7,15)(10,20)(13,28)(14,22)(16,26)(17,27)(18,23)(19,25)(21,29)(24,31)(30,32)(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)(2,7)(4,17)(5,14)(6,16)(8,21)(9,12)(10,26)(11,22)(13,19)(15,20)(18,30)(23,27)(24,28)(25,32)(29,31)(34,40)(35,46)(36,49)(37,38)(42,57)(43,44)(45,53)(48,52)(50,51)(55,61)(59,64)(60,62);(1,4,6,19)(2,8,11,25)(3,13,16,17)(5,18,20,31)(7,21,22,32)(9,23,10,24)(12,27,26,28)(14,29,15,30)(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);(2,9)(3,14)(5,20)(8,23)(10,11)(12,26)(13,29)(15,16)(17,30)(18,31)(24,25)(27,28)(33,36)(34,42)(35,47)(37,50)(38,49)(39,51)(40,54)(41,55)(43,58)(44,57)(45,59)(46,60)(48,63)(52,62)(53,61)(56,64)

Polycyclic presentation:

2^9|c^2=hi,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, 239131);