#1907212

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1606 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1814408);