#332018

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

489 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,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=ghi,f^c=fi,g^c=ghi,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, 239214);