#7300535

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7207731);