#4973832

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

1588 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)(3,4,8,7)(10,11)(14,15)(17,18,22,20)(19,24,21,23)(25,26,30,31)(27,29,35,32)(28,33,37,38)(34,36,40,39);(9,10,13,11)(12,15,16,14)(26,31)(29,32)(33,38)(36,39);(2,6)(3,8)(10,11)(14,15)(17,19,22,21)(18,23,20,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,38)(37,40);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24)(25,28,30,37)(26,33,31,38)(27,34,35,40)(29,36,32,39);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24)(25,29,30,32)(26,27,31,35)(28,36,37,39)(33,34,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fgi,d^2=gh,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, 4881028);