#408609

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di

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, 315805);