#536695

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2482 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)(10,13)(11,15)(17,18,23,26)(19,22,20,24)(21,25,30,32)(27,29,28,31);(2,4)(10,13)(14,16)(17,19,23,20)(18,24,26,22)(21,27,30,28)(25,31,32,29);(1,2,3,4)(5,7,8,6)(9,10,12,13)(11,14,15,16)(17,20,23,19)(18,24,26,22)(21,27,30,28)(25,29,32,31);(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(17,22,23,24)(18,19,26,20)(21,29,30,31)(25,27,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443891);