#496830

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,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, 404026);