#380144

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1334 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:

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

Polycyclic presentation:

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

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, 287340);