#431957

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ef,d^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=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, 339153);