#410602

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317798);