#381395

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=egi,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de

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, 288591);