#232873

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	#60967	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1119 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)(35,45)(36,50)(37,43)(38,42)(39,44)(40,48)(41,51)(46,54)(47,56)(49,55)(52,59)(53,58)(57,62)(60,61)(63,64);(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)(33,35,39,49)(34,40,44,56)(36,46,53,62)(37,47,38,48)(41,57,59,54)(42,55,43,45)(50,60,58,64)(51,61,52,63);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,25)(11,26)(12,29)(14,17)(15,31)(16,30)(20,27)(23,32)(24,28)(33,36)(34,41)(35,46)(37,51)(38,52)(39,53)(40,54)(42,50)(43,58)(44,59)(45,60)(47,61)(48,63)(49,62)(55,64)(56,57);(2,10)(8,23)(9,25)(11,20)(12,28)(22,32)(24,29)(26,27)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,53,52)(40,55,56,45)(41,50,59,58)(46,61,62,63)(54,64,57,60)

Polycyclic presentation:

2^9|b^2=h,d^2=i,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, 140069);