#2826980

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1710 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fg,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2734176);