#7016725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,37,22,28,21,27)(19,29,31,41,32,24,33,40)(20,26,34,25,35,42,36,43)(30,45,48,44,47,39,46,38);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,28,29)(20,30,35,47)(21,33,23,31)(25,38,43,44)(26,39,42,45)(27,41,37,40)(34,48,36,46);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,37)(27,42)(28,43)(29,44)(31,46)(32,47)(33,48)(39,40)(41,45);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,39)(25,37)(27,43)(28,42)(29,45)(31,48)(32,47)(33,46)(38,40)(41,44);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,23)(18,27,28,37)(19,31,32,33)(20,34,35,36)(24,40,29,41)(25,42,43,26)(30,46,47,48)(38,39,44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6923921);