#6546242

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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:

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

Polycyclic presentation:

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

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, 6453438);