#380969

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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)(21,22,26,28,25,29,24,30)(23,31,34,36,33,27,32,35);(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,23,25,33)(22,27,29,31)(24,32,26,34)(28,35,30,36);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(22,28)(24,26)(27,35)(29,30)(31,36)(32,34);(7,11)(10,12)(21,24,25,26)(22,28,29,30)(23,32,33,34)(27,35,31,36)

Polycyclic presentation:

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

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, 288165);