#3036025

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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

1696 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)(3,6)(4,7)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,39)(38,40);(1,3,4,8)(2,5,7,6)(17,18,20,22)(19,21,24,23)(26,27)(28,35)(29,37)(39,40);(9,11,13,12)(10,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,40,38,39);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19,20,24)(18,21,22,23)(25,28)(26,31)(27,33)(29,38)(30,35)(32,40)(34,39)(36,37);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,16,14)(17,18)(19,23)(20,22)(21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=ghi,d^2=gh,e^2=hi,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, 2943221);