#463246

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 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)(3,4)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,28)(20,31)(21,26)(22,25)(23,27)(24,29)(30,32);(9,11,13,16)(10,14,12,15)(17,19,21,29,23,20,22,30)(18,24,25,28,27,32,26,31);(5,6,8,7)(10,12)(14,15)(18,25)(19,29)(20,30)(21,22)(24,32)(26,27);(2,4)(5,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,21,30,23,19,22,29)(18,24,25,28,27,32,26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370442);