#6995445

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 6902641);