#2526544

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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fghi,c^2=gh,d^2=h,e^2=hi,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, 2433740);