#7184565

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,26,22,27,34,23)(19,24,30,39,29,38,44,35)(20,25,31,42,33,41,46,36)(28,37,43,48,32,40,45,47);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(18,23)(20,31)(21,34)(24,35)(26,27)(28,43)(30,44)(32,45)(33,46)(36,42)(38,39)(47,48);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,24,27,38)(20,32,33,28)(21,30,34,44)(23,35,26,39)(25,40,41,37)(31,45,46,43)(36,48,42,47);(10,13)(14,16)(17,20,21,31,22,33,34,46)(18,25,26,42,27,41,23,36)(19,28,30,43,29,32,44,45)(24,37,39,48,38,40,35,47);(9,11,12,15)(10,14,13,16)(19,29)(20,33)(24,38)(25,41)(30,44)(31,46)(35,39)(36,42)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7091761);