#441210

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1538 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,23,29,25,30,26,27)(24,32,33,36,34,35,31,28);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20)(21,23,25,26)(22,27,30,29)(24,33,34,31)(28,35,36,32);(2,4)(5,6,9,8)(7,10,12,11)(15,19)(17,20)(21,24,25,34)(22,28,30,36)(23,31,26,33)(27,32,29,35);(5,8,9,6)(7,10,12,11)(22,29)(23,26)(27,30)(28,35)(31,33)(32,36)

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=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, 348406);