#545510

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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

2406 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)(9,10,12,14)(11,13,16,15)(17,18,23,26)(19,21,20,24)(22,30,31,25)(27,32,28,29);(2,4)(9,10,12,14)(11,15,16,13)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(1,2,3,4)(5,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,21,26,24)(22,28,31,27)(25,29,30,32);(9,11,12,16)(10,13,14,15)(17,21)(18,20)(19,26)(22,29)(23,24)(25,28)(27,30)(31,32);(17,22,23,31)(18,25,26,30)(19,27,20,28)(21,29,24,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=fg,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, 452706);