#359306

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1412 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266502);