#360713

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1718 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267909);