#544553

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(5,6,7,8)(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(2,4)(10,14)(11,17)(16,20)(19,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,3,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=fg,e^2=g,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, 451749);