#429931

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
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

1036 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)(5,6,9,11)(7,10,8,12)(13,14,17,23,19,25,18,24)(15,26,29,39,31,38,30,20)(16,21,33,22,35,42,34,41)(27,37,44,43,28,40,32,36);(5,7,9,8)(6,10,11,12)(13,15,19,31)(14,20,25,39)(16,32,35,44)(17,30,18,29)(21,40,42,37)(22,43,41,36)(23,38,24,26)(27,33,28,34);(6,11)(10,12)(13,16)(14,21)(15,27)(17,33)(18,34)(19,35)(20,36)(22,23)(24,41)(25,42)(26,37)(28,31)(29,44)(30,32)(38,40)(39,43);(2,4)(5,8,9,7)(6,12,11,10)(13,16)(14,22)(15,28)(17,34)(18,33)(19,35)(20,37)(21,23)(24,42)(25,41)(26,36)(27,31)(29,44)(30,32)(38,43)(39,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,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, 337127);