#6830071

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,28)(26,31)(30,32);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,25)(21,30)(22,28)(23,31)(26,32)(27,29);(10,13)(14,16)(17,20,23,29,22,24,30,26)(18,19,27,21,25,28,32,31);(2,6)(5,7)(18,25)(20,24)(26,29)(27,32);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,28,22,31,30,19)(18,26,27,20,25,29,32,24)

Polycyclic presentation:

2^9|b^2=fi,c^2=h,e^2=gh,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, 6737267);