#7300493

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=f,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=eh,d^c=dh,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, 7207689);