#353202

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂²	C₂×C₄³
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=f,f^2=i,b^a=be,c^a=ci,d^a=di,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, 260398);