#414633

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 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)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,24,29,32)(20,26,21,25)(23,30,31,28);(5,6)(7,8)(17,19,22,29)(18,23,27,31)(20,28,21,30)(24,26,32,25);(2,4)(6,8)(17,19)(18,24)(20,30)(21,28)(22,29)(23,25)(26,31)(27,32);(9,11,12,16)(10,13,14,15)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,24,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,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, 321829);