#545488

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 452684);