#2548608

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

1742 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=ghi,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, 2455804);