#6437257

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,32)(24,35)(25,31)(26,34)(27,29)(28,30)(33,36);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,26,30,27,33,25,35)(22,28,31,32,29,24,34,36);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(22,29)(23,33)(30,35)(31,34);(6,11)(10,16)(12,18)(17,20)(21,24)(22,30)(23,34)(25,32)(26,36)(27,28)(29,35)(31,33);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(21,25,27,26)(22,31,29,34)(23,35,33,30)(24,36,28,32)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^d=ei

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, 6344453);