#7008123

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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)(17,18,21,26,22,25,36,42)(19,27,30,37,28,43,35,23)(20,31,34,47,33,24,39,41)(29,45,46,38,32,44,48,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(19,28)(23,37)(27,43)(29,32)(30,35)(38,40)(44,45)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,25,37)(20,32,33,29)(21,35,36,30)(24,40,31,38)(26,43,42,27)(34,46,39,48)(41,44,47,45);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20)(18,24)(19,29)(21,34)(22,33)(23,38)(25,31)(26,41)(27,44)(28,32)(30,46)(35,48)(36,39)(37,40)(42,47)(43,45);(10,14)(13,15)(18,25)(20,33)(23,37)(26,42)(27,43)(29,32)(34,39)(46,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=f,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, 6915319);