#1151999

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1814 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fghi,d^2=fgh,e^2=f,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, 1059195);