#7052135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6959331);