#6498658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1838 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)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,25,28,26,30,23,27)(24,29,33,35,32,36,31,34);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(22,27)(23,25)(24,32)(28,30)(29,35)(34,36);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,23,26,25)(22,28,30,27)(24,31,32,33)(29,35,36,34);(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,31)(25,33)(26,32)(27,34)(28,35)(30,36);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,30)(24,32)(27,28)(31,33)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,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, 6405854);