#410660

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1168 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)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,20,25,21,24,28,22)(19,23,26,30,27,31,32,29);(2,4)(5,6,8,7)(10,14)(15,16)(18,22)(19,26)(20,28)(24,25)(27,32)(29,30);(5,6,8,7)(9,11,13,12)(10,15,14,16)(17,19,20,26,21,27,28,32)(18,23,25,30,24,31,22,29);(5,7,8,6)(9,12,13,11)(10,15,14,16)(18,24)(19,27)(22,25)(26,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=hi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,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, 317856);