#7155665

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

1547 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)(11,15)(14,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,40)(29,38)(31,45)(33,46)(34,47)(35,42)(36,48)(39,44)(41,43);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,46)(21,34,36,43,35,31,29,44)(26,41,38,47,42,39,48,45);(3,8)(6,7)(10,13)(14,16)(17,21)(18,26)(19,29)(20,31)(22,36)(23,35)(24,38)(25,39)(27,42)(28,43)(30,45)(32,44)(33,34)(37,48)(40,41)(46,47);(9,11,12,15)(10,14,13,16)(21,35)(26,42)(29,36)(31,34)(38,48)(39,41)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,e^2=h,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, 7062861);