#2411385

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1696 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)(3,8)(4,7)(5,6)(9,10,12,13)(11,15,16,14)(17,18,21,23)(19,24,20,22)(25,26,30,32)(27,33,34,28)(29,37,38,31)(35,36,40,39);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23)(27,34)(28,33)(35,40)(36,39);(2,5)(7,8)(10,13)(14,15)(17,19,21,20)(18,22,23,24)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,39)(38,40);(1,4,6,3)(2,8,5,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,19)(18,22,23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23)(25,29,30,38)(26,31,32,37)(27,35,34,40)(28,36,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fi,d^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2318581);