#471128

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,26,31,27)(28,30,32,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,26,29,32,31,27,30,28);(3,7)(6,8)(10,15)(14,16)(19,23)(21,24)(25,27,31,26)(28,29,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(25,28,29,26,31,32,30,27)

Polycyclic presentation:

2^9|a^2=egi,b^2=fh,c^2=i,d^2=ef,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=di,d^c=de,e^c=ei,g^c=gi,h^c=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, 378324);