#6844319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42541	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=gi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6751515);