#3034138

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 2097152
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

1822 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=ghi,d^2=g,e^2=ghi,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, 2941334);