#450245

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1116 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:

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

Polycyclic presentation:

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

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, 357441);