#2260778

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1712 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,16)(14,15)(18,19)(20,24)(25,26,29,30)(27,31,33,37)(28,32,36,38)(34,39,35,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,18,21,19)(20,22,24,23)(26,30)(27,33)(32,38)(34,35);(2,6)(4,7)(9,12,14,13)(10,11,15,16)(17,19,21,18)(20,22,24,23)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,3)(2,4)(5,8)(6,7)(9,13,14,12)(10,11,15,16)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,22)(19,23)(21,24)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=fi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2167974);