#232728

order	512
nilpotency class	3
isoclinism	isoclinic class of #232724
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	#60948	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1121 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)(4,9)(5,10)(6,11)(7,12)(8,15)(14,29)(16,24)(17,23)(18,22)(19,26)(20,28)(21,27)(25,30)(31,32)(33,34)(35,39)(36,46)(37,49)(38,42)(40,48)(41,51)(43,58)(44,53)(45,62)(47,57)(50,56)(52,60)(54,61)(55,63)(59,64);(1,3,7,17)(2,8,12,24)(4,18,20,31)(5,16,6,15)(9,25,28,29)(10,13,11,23)(14,21,30,19)(22,27,32,26)(33,35)(34,39)(36,43)(37,45)(38,44)(40,55)(41,54)(42,53)(46,64)(47,59)(48,60)(49,62)(50,63)(51,61)(52,56)(57,58);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,27)(12,28)(13,29)(15,18)(16,31)(17,30)(23,25)(24,32)(33,36)(34,40)(35,43)(37,50)(38,47)(39,52)(41,57)(42,56)(44,59)(45,63)(46,49)(48,51)(53,55)(54,64)(58,62)(60,61);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,26,28,27)(14,18,30,31)(22,25,32,29)(35,44)(39,53)(43,59)(45,61)(52,55)(54,62)(58,64)(60,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139924);