#232831

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	#60961	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1110 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)(35,40)(36,48)(37,42)(38,43)(39,44)(41,52)(45,58)(46,55)(47,57)(49,54)(50,59)(51,60)(53,56)(61,63)(62,64);(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)(41,58)(42,55)(43,56)(44,57)(45,50)(48,63)(51,61)(52,62)(54,60)(59,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,18)(12,27)(13,28)(15,29)(16,31)(17,30)(23,25)(24,32)(33,36)(34,41)(35,45)(37,50)(38,52)(39,51)(40,54)(42,48)(43,60)(44,59)(46,61)(47,62)(49,53)(55,58)(56,64)(57,63);(2,10)(8,23)(9,26)(11,12)(13,24)(18,27)(22,25)(28,32)(33,37,39,38)(34,42,44,43)(35,46,47,53)(36,50,51,52)(40,55,57,56)(41,48,59,60)(45,61,62,49)(54,58,63,64)

Polycyclic presentation:

2^9|b^2=h,d^2=i,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, 140027);