#2044400

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

1596 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(25,26,29,30)(27,33,34,31)(28,32,36,37)(35,39,40,38);(2,5)(3,7)(10,12)(14,15)(17,18,21,20)(19,23,24,22)(26,30)(27,34)(28,36)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,6,8)(2,7,5,3)(17,20,21,18)(19,23,24,22)(25,27,29,34)(26,31,30,33)(28,35,36,40)(32,38,37,39);(1,5,6,2)(3,8,7,4)(9,12,13,10)(11,15,16,14)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fh,d^2=h,e^2=fg,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, 1951596);