#6997013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,30,42)(20,24,32,40)(21,34,36,25)(23,35,38,29)(28,43,31,45)(33,48,47,41)(37,46,39,44);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,26,38)(20,31,32,28)(21,35,36,29)(24,39,40,37)(25,42,34,27)(33,44,47,46)(41,43,48,45);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,20,22,32)(18,24,26,40)(19,28,30,31)(21,33,36,47)(23,37,38,39)(25,41,34,48)(27,43,42,45)(29,44,35,46);(10,14)(13,15)(20,32)(24,40)(28,31)(33,47)(37,39)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,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,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6904209);