#434382

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341578);