#2608148

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2193 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2515344);