#6818254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,e^2=fh,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^c=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, 6725450);