#4267784

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1578 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fgh,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4174980);