#232828

order	512
nilpotency class	3
isoclinism	isoclinic class of #232820
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1086 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)(3,13)(4,18)(5,12)(6,11)(7,10)(8,15)(9,19)(14,25)(16,23)(17,24)(20,27)(21,26)(22,30)(28,29)(31,32)(33,34,39,44)(35,40,47,56)(36,48,50,59)(37,43,38,42)(41,52,58,51)(45,57,61,64)(46,60,53,55)(49,54,62,63);(1,3,7,17)(2,8,10,23)(4,14,20,30)(5,16,6,15)(9,25,26,22)(11,24,12,13)(18,28,27,32)(19,31,21,29)(33,35)(34,40)(36,49)(37,46)(38,53)(39,47)(41,57)(42,55)(43,60)(44,56)(45,52)(48,63)(50,62)(51,61)(54,59)(58,64);(1,4,7,20)(2,9,10,26)(3,14,17,30)(5,19,6,21)(8,22,23,25)(11,18,12,27)(13,28,24,32)(15,29,16,31)(33,36)(34,41)(35,45)(37,52)(38,51)(39,50)(40,54)(42,59)(43,48)(44,58)(46,62)(47,61)(49,53)(55,64)(56,63)(57,60);(2,10)(8,23)(9,26)(11,12)(13,24)(18,27)(22,25)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140024);