#2076365

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

1649 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fhi,d^2=fh,e^2=fghi,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, 1983561);