#425639

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,e^d=ei,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, 332835);