#7007627

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(17,18,23,37,22,26,21,27)(19,28,36,24,29,43,31,39)(20,32,38,42,34,25,35,47)(30,45,48,41,33,44,46,40);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(19,29)(24,39)(28,43)(30,33)(31,36)(40,41)(44,45)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,26,39)(20,33,34,30)(21,36,23,31)(25,41,32,40)(27,43,37,28)(35,46,38,48)(42,44,47,45);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,34)(23,38)(24,40)(26,32)(27,42)(28,44)(29,33)(31,46)(36,48)(37,47)(39,41)(43,45);(10,14)(13,15)(18,26)(20,34)(24,39)(27,37)(28,43)(30,33)(35,38)(46,48)

Polycyclic presentation:

2^9|a^2=gi,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, 6914823);