#432115

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1017 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)(5,6,9,12)(7,10,8,11)(13,14,19,20,18,24,17,23)(15,25,33,36,28,21,26,34)(16,29,32,22,31,38,30,37)(27,41,40,39,43,44,42,35);(5,7,9,8)(6,10,12,11)(14,20)(15,26)(16,30)(17,19)(22,37)(23,24)(28,33)(31,32)(34,36)(35,44)(39,41)(40,42);(5,7,9,8)(6,11,12,10)(13,15,18,28)(14,21,24,25)(16,27,31,43)(17,26,19,33)(20,34,23,36)(22,35,37,39)(29,44,38,41)(30,42,32,40);(2,4)(5,8,9,7)(6,11,12,10)(13,16)(14,22)(15,27)(17,32)(18,31)(19,30)(20,29)(21,35)(23,38)(24,37)(25,39)(26,40)(28,43)(33,42)(34,44)(36,41)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=f,d^2=fi,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, 339311);