#6393908

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#537	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2997 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,27,31,26,30,25,36)(23,24,33,35,32,28,29,34);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23)(22,28)(24,30)(25,29)(26,32)(27,33)(31,34)(35,36);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,27,35,26,28,25,34)(22,29,31,23,30,33,36,32);(7,16)(11,14)(15,17)(19,20)(22,30)(23,32)(29,33)(31,36);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(21,25,26,27)(22,31,30,36)(23,33,32,29)(24,34,28,35)

Polycyclic presentation:

2^9|a^2=h,c^2=fgh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301104);