#250486

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157682);