#429009

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1027 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ei,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336205);