#497726

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×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 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)(7,9)(8,10)(11,12)(13,14,17,22)(15,24,26,37)(16,19,31,35)(18,32,30,23)(20,33,38,27)(21,39,36,28)(25,41,42,34)(29,40,43,44);(5,7,8,11)(6,9,10,12)(14,19)(15,25)(16,28)(18,30)(20,37)(21,23)(22,35)(24,38)(26,42)(27,43)(29,33)(31,39)(32,36)(40,44);(6,10)(9,12)(15,26)(20,38)(24,37)(25,42)(27,33)(29,43)(34,41)(40,44);(13,15,17,26)(14,20,22,38)(16,29,31,43)(18,33,30,27)(19,34,35,41)(21,40,36,44)(23,37,32,24)(25,39,42,28)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,d^c=di,e^d=eh,h^d=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, 404922);