#7004490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1733 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)(11,13)(12,14)(15,16)(17,18,22,27,23,28,38,45)(19,29,33,41,32,24,31,42)(20,26,35,43,21,36,34,25)(30,40,47,39,37,44,48,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,37)(22,33)(23,32)(25,39)(26,44)(27,42)(28,29)(31,38)(34,48)(35,47)(36,40)(41,45)(43,46);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,25,28,43)(19,30,32,37)(22,34,38,35)(24,39,29,46)(26,27,36,45)(31,47,33,48)(40,41,44,42);(3,8)(6,7)(11,16)(13,15)(17,21,23,20)(18,26,28,36)(19,30,32,37)(22,34,38,35)(24,40,29,44)(25,27,43,45)(31,48,33,47)(39,41,46,42);(10,14)(13,15)(17,22,23,38)(18,27,28,45)(19,31,32,33)(20,34,21,35)(24,41,29,42)(25,36,43,26)(30,47,37,48)(39,44,46,40)

Polycyclic presentation:

2^9|a^2=fgi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6911686);