#355790

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

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

1074 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)(11,14)(12,13)(15,16)(17,18,20,25,22,27,23,28)(19,29,21,26,30,24,31,32);(10,13)(11,15)(17,19,23,31,22,30,20,21)(18,24,28,26,27,29,25,32);(2,4)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,25,27,28)(19,21,30,31)(24,32,29,26);(5,7,8,6)(10,13)(14,16)(17,21,20,30,22,31,23,19)(18,26,25,24,27,32,28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei

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, 262986);