#3856884

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

1772 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=fghi,d^2=ghi,e^2=ghi,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, 3764080);