#6914685

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#30513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,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, 6821881);