#4416336

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

1842 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)(3,7)(10,11)(12,16)(17,18)(19,20)(21,24)(22,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,24)(21,23)(27,33)(31,34)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,11,13,10)(12,14,16,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,24)(21,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4323532);