#1642359

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1894 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)(10,12)(11,15)(19,20)(22,23)(25,26)(27,33)(28,36)(29,35)(30,31)(32,37)(34,39)(38,40);(1,2,4,6)(3,5,8,7)(11,15)(14,16)(17,18,21,24)(19,22,20,23)(26,31)(28,37)(29,33)(34,38);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,27)(26,29)(28,38)(30,35)(31,33)(32,40)(34,37)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,39)(30,37)(31,36)(33,40)(35,38);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,29)(26,27)(28,39)(30,33)(31,35)(32,34)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=fgh,e^2=fgh,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, 1549555);