#356558

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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)(10,14)(15,16)(17,18,23,26,22,25,20,21)(19,27,28,32,30,24,31,29);(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,31)(21,32)(22,30)(23,28)(25,27)(26,29);(2,4)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,21,25,26)(19,28,30,31)(24,29,27,32);(9,12,13,11)(10,15,14,16)(17,21,20,25,22,26,23,18)(19,29,31,24,30,32,28,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=eg,g^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263754);