#6386945

order	512
nilpotency class	3
isoclinism	isoclinic class of #62253
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6294141);