#2071404

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

1492 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fhi,d^2=ghi,e^2=f,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, 1978600);