#4416567

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=gh,e^2=fg,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, 4323763);