#7023176

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

1593 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,22,27)(20,31,34,39)(21,26,37,43)(25,29,41,32)(30,38,36,44)(33,47,35,48)(40,45,42,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,28,27)(20,32,34,29)(21,30,37,36)(25,39,41,31)(26,38,43,44)(33,46,35,45)(40,48,42,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,32,23,34,19,29)(18,25,24,31,28,41,27,39)(21,35,36,45,37,33,30,46)(26,42,38,48,43,40,44,47);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,21)(18,26)(19,30)(20,33)(22,36)(23,37)(24,38)(25,40)(27,44)(28,43)(29,45)(31,47)(32,46)(34,35)(39,48)(41,42);(10,14)(13,15)(17,19,23,22)(18,27,28,24)(20,32,34,29)(21,36,37,30)(25,31,41,39)(26,38,43,44)(33,45,35,46)(40,48,42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=f,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=ehi,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, 6930372);