#380461

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22,26,30,25,29,24,28)(23,31,34,35,33,27,32,36);(1,2)(3,4)(6,9)(10,12)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,31)(30,36);(2,4)(5,7,8,11)(6,10,9,12)(14,17)(18,20)(21,24,25,26)(22,28,29,30)(23,32,33,34)(27,35,31,36);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(23,33)(27,31)(32,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287657);