#420664

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327860);