#4975981

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1608 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)(5,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,18,20,22)(19,21,23,24)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40);(1,3,6,5)(2,4,7,8)(17,19,20,23)(18,21,22,24)(26,31)(27,33)(29,37)(35,40);(2,7)(4,8)(9,11,14,13)(10,15,12,16)(17,19)(18,21)(20,23)(22,24)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(1,4,6,8)(2,3,7,5)(9,12,14,10)(11,15,13,16)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,5,6,3)(2,8,7,4)(9,13,14,11)(10,15,12,16)(17,18)(19,21)(20,22)(23,24)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fghi,c^2=fi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4883177);