#6389939

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^c=ei

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, 6297135);