#496828

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

734 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)(7,11)(8,10)(9,12)(13,14,18,24)(15,20,27,37)(16,22,19,34)(17,30,32,44)(21,39,40,26)(23,29,42,33)(25,35,28,38)(31,36,43,41);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,28)(17,26)(18,27)(19,25)(21,30)(22,38)(23,36)(24,37)(29,43)(31,33)(32,39)(34,35)(40,44)(41,42);(7,12)(9,11)(13,15)(14,21)(16,25)(17,31)(18,27)(19,28)(20,30)(22,36)(23,38)(24,40)(26,33)(29,39)(32,43)(34,41)(35,42)(37,44);(6,10)(9,11)(13,16,18,19)(14,22,24,34)(15,25,27,28)(17,29,32,33)(20,35,37,38)(21,36,40,41)(23,30,42,44)(26,31,39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404024);