#7187534

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=fg,d^2=fi,e^2=hi,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=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, 7094730);