#7227798

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1148 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,24)(19,22,25,30)(20,27,29,23)(26,32,28,31)(33,34,40,45)(35,41,49,59)(36,43,37,51)(38,44,39,54)(42,52,61,50)(46,56,47,62)(48,57,53,58)(55,63,60,64);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19)(18,22)(20,28)(21,25)(23,32)(24,30)(26,29)(27,31)(33,35,40,49)(34,41,45,59)(36,47,37,46)(38,53,39,48)(42,60,61,55)(43,62,51,56)(44,58,54,57)(50,63,52,64);(19,25)(22,30)(26,28)(31,32)(33,36,40,37)(34,42,45,61)(35,46,49,47)(38,50,39,52)(41,55,59,60)(43,44,51,54)(48,63,53,64)(56,58,62,57);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,29)(18,23,24,27)(19,26,25,28)(22,31,30,32)(33,37,40,36)(34,43,45,51)(35,47,49,46)(38,50,39,52)(41,56,59,62)(42,44,61,54)(48,63,53,64)(55,58,60,57);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(19,25)(22,30)(26,28)(31,32)(33,38,40,39)(34,44,45,54)(35,48,49,53)(36,50,37,52)(41,57,59,58)(42,51,61,43)(46,63,47,64)(55,56,60,62)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7134994);