#4935893

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2224 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=h,d^2=h,e^2=f,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, 4843089);