#356543

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
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

1130 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)(17,18,21,20,22,24,30,29)(19,23,27,28,25,31,32,26);(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,28)(21,27)(22,25)(24,31)(26,29)(30,32);(2,4)(9,11,13,12)(10,15,14,16)(19,25)(23,31)(26,28)(27,32);(9,12,13,11)(10,15,14,16)(17,20,21,24,22,29,30,18)(19,26,27,23,25,28,32,31)

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,d^a=di,c^b=ci,d^b=dei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263739);