#2063755

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1434 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fghi,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1970951);