#7117826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1429 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=ghi,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7025022);