#7023179

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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)(17,18,22,26)(19,23,30,27)(20,31,34,37)(21,25,36,43)(24,32,40,28)(29,38,46,42)(33,47,35,48)(39,45,41,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,46)(24,31,40,37)(25,38,43,42)(33,45,35,44)(39,47,41,48);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,37,26,40,27,31)(21,35,46,44,36,33,29,45)(25,41,38,47,43,39,42,48);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,21)(18,25)(19,29)(20,33)(22,36)(23,38)(24,39)(26,43)(27,42)(28,44)(30,46)(31,47)(32,45)(34,35)(37,48)(40,41);(10,14)(13,15)(18,26)(20,34)(21,36)(23,27)(28,32)(29,46)(39,41)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^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=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, 6930375);