#498063

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

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 405259);