#426032

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

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

Polycyclic presentation:

2^9|a^2=ei,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, 333228);