#7598666

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1659 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7505862);