#6818177

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725373);