#497715

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

710 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)(15,23)(16,22)(17,31)(19,20)(21,26)(24,36)(25,37)(27,40)(29,30)(32,34)(35,39)(38,44)(41,43);(5,7,8,11)(6,9,10,12)(13,15,20,25)(14,21,19,34)(16,24,30,38)(17,32,31,26)(18,23,33,37)(22,35,29,43)(27,41,40,39)(28,36,42,44);(6,10)(9,12)(13,16,20,30)(14,22,19,29)(15,24,25,38)(17,27,31,40)(18,28,33,42)(21,35,34,43)(23,36,37,44)(26,39,32,41);(13,17)(14,18)(15,25)(16,27)(19,33)(20,31)(22,28)(23,32)(24,38)(26,37)(29,42)(30,40)(36,41)(39,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^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,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, 404911);