#2804842

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 2097152
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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=fh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

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, 2712038);