#426751

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

1062 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)(6,9)(10,12)(13,14,17,22,18,23,33,20)(15,24,26,36,27,39,32,19)(16,28,30,38,31,40,34,21)(25,35,29,37,42,44,43,41);(5,6,8,9)(7,10,11,12)(13,15,18,27)(14,19,23,36)(16,29,31,43)(17,32,33,26)(20,24,22,39)(21,37,38,41)(25,34,42,30)(28,35,40,44);(5,7,8,11)(6,10,9,12)(14,20)(17,33)(19,24)(21,28)(22,23)(26,32)(29,43)(30,34)(35,41)(36,39)(37,44)(38,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,34)(18,31)(19,35)(20,28)(22,40)(23,38)(24,41)(26,43)(27,42)(29,32)(30,33)(36,44)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 333947);