#6841215

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,32)(29,30);(1,2,4,5)(3,6,7,8)(18,23)(19,27)(20,29)(22,28)(24,25)(30,31);(10,14)(15,16)(17,19,28,20,21,29,22,27)(18,24,32,25,23,30,26,31);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(18,23)(19,29)(20,27)(26,32);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,20,22,19,21,27,28,29)(18,25,26,24,23,31,32,30)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6748411);