#4631715

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1780 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)(3,8,7,4)(9,10,14,15)(11,16,12,13)(17,18,21,23)(19,24,20,22)(25,26)(27,31)(28,30)(29,36)(32,35)(33,37)(34,39)(38,40);(2,6)(3,7)(10,15)(11,12)(17,18)(19,22)(20,24)(21,23)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(2,6)(4,8)(9,11,14,12)(10,13,15,16)(17,19,21,20)(18,22,23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,13,15,16)(17,20,21,19)(18,22,23,24)(25,28,30,26)(27,31,35,32)(29,36,37,33)(34,38,40,39);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,24)(20,22)(21,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4538911);