#473667

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

864 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380863);