#6990000

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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=gi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,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, 6897196);