#550053

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

2248 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,d^2=g,e^2=i,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, 457249);