#6458968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 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)(11,13,15,16)(18,23)(19,26)(20,21)(24,31)(25,29)(30,32);(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,24,26,32)(20,23,21,29)(27,30,28,31);(2,5)(6,8)(11,15)(13,16)(17,19,22,26)(18,24,25,32)(20,27,21,28)(23,30,29,31);(11,15)(13,16)(19,26)(24,32)(27,28)(30,31);(1,3,4,7)(2,6,5,8)(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,30,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366164);