#7019622

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,e^2=i,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, 6926818);