#494793

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401989);