#496353

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403549);