#7300746

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,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, 7207942);