#355403

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×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

1118 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)(10,13)(11,15)(17,18,22,20,21,25,23,26)(19,24,29,31,28,32,30,27);(9,10)(11,14)(12,13)(15,16)(17,19,23,30,21,28,22,29)(18,24,26,27,25,32,20,31);(2,4)(5,7,8,6)(10,13)(14,16)(17,20,22,25,21,26,23,18)(19,27,29,24,28,31,30,32);(9,11,12,15)(10,14,13,16)(19,28)(24,32)(27,31)(29,30)

Polycyclic presentation:

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

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, 262599);