#4957791

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1510 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4864987);