#6546283

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,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, 6453479);