#1849458

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1450 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=gh,d^2=fghi,e^2=fh,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, 1756654);