#498059

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,19,25,28,40,36,44,18,22,17,24,20,26,33,41)(15,27,30,23,34,43,35,21,29,39,16,32,31,42,37,38);(5,7,9,8)(6,10,11,12)(13,15,18,29)(14,21,22,38)(16,33,30,36)(17,35,19,37)(20,34,28,31)(23,26,32,40)(24,42,25,43)(27,41,39,44);(5,8,9,7)(6,10,11,12)(14,22)(21,38)(23,32)(24,25)(26,40)(27,39)(41,44)(42,43);(13,16,33,29,20,35,17,34,18,30,36,15,28,37,19,31)(14,23,41,27,26,38,24,42,22,32,44,39,40,21,25,43)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405255);