#545390

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^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, 452586);