#232870

order	512
nilpotency class	3
isoclinism	isoclinic class of #232862
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

1096 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)(3,12)(4,9)(5,20)(6,11)(7,10)(8,14)(13,29)(15,23)(16,28)(17,22)(18,27)(19,25)(21,26)(24,30)(31,32)(33,34,39,42)(35,45,48,59)(36,49,52,58)(37,44,38,43)(40,53,55,47)(41,50,57,51)(46,54,62,56)(60,63,64,61);(1,3,7,16)(2,8,10,23)(4,17,19,31)(5,15,6,14)(9,24,25,29)(11,28,20,12)(13,21,30,18)(22,26,32,27)(33,35,39,48)(34,40,42,55)(36,46,52,62)(37,53,38,47)(41,56,57,54)(43,59,44,45)(49,60,58,64)(50,63,51,61);(1,4,7,19)(2,9,10,25)(3,13,16,30)(5,18,6,21)(8,22,23,32)(11,26,20,27)(12,29,28,24)(14,17,15,31)(33,36,39,52)(34,41,42,57)(35,46,48,62)(37,51,38,50)(40,54,55,56)(43,49,44,58)(45,60,59,64)(47,63,53,61);(2,10)(8,23)(9,25)(11,20)(12,28)(22,32)(24,29)(26,27)(34,42)(40,55)(41,57)(43,44)(45,59)(49,58)(54,56)(60,64)

Polycyclic presentation:

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

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, 140066);