#611359

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

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, 518555);