#356914

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

1162 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=fg,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=dei,d^c=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, 264110);