#1189907

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1097103);