#365949

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1430 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:

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

Polycyclic presentation:

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

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, 273145);