#6830195

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	#42971	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 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)(17,18,22,23)(19,24,28,32)(20,27,29,25)(21,26,30,31);(1,3,4,8)(2,5,6,7)(17,19,22,28)(18,24,23,32)(20,21,29,30)(25,31,27,26);(10,13)(14,16)(17,20,18,25,22,29,23,27)(19,26,32,30,28,31,24,21);(2,6)(5,7)(17,18,22,23)(19,24,28,32)(20,27,29,25)(21,26,30,31);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,18,26,22,30,23,31)(19,27,32,20,28,25,24,29)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=i,e^2=gh,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, 6737391);