#353759

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260955);