#369385

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1558 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)(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,25,31)(20,30,32,29)(21,26,22,28);(1,2)(3,4)(5,7,8,6)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(2,4)(9,11,13,12)(10,15,16,14)(17,20,22,24,23,32,21,31)(18,25,26,30,27,19,28,29);(9,12,13,11)(10,15,16,14)(17,21,23,22)(18,26,27,28)(19,29,25,30)(20,31,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276581);