#7117811

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(17,18,23,24,22,28,21,27)(19,25,35,37,30,40,31,41)(20,32,36,42,34,26,33,38)(29,43,46,48,45,39,44,47);(10,13)(11,15)(18,24)(20,33)(21,23)(25,37)(27,28)(29,44)(31,35)(34,36)(38,42)(40,41)(45,46)(47,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,29)(21,31)(22,30)(23,35)(24,37)(26,39)(27,41)(28,40)(32,43)(33,44)(34,45)(36,46)(38,47)(42,48);(10,13)(14,16)(17,20,21,33,22,34,23,36)(18,26,27,42,28,32,24,38)(19,29,31,44,30,45,35,46)(25,39,41,48,40,43,37,47);(1,4,5,3)(2,6,7,8)(9,11,12,15)(10,14,13,16)(19,30)(20,34)(25,40)(26,32)(31,35)(33,36)(37,41)(38,42)

Polycyclic presentation:

2^9|a^2=fghi,c^2=gh,d^2=f,e^2=g,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, 7025007);