#3396088

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1736 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=gh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3303284);