#4944531

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,18)(20,26)(21,28)(22,23)(25,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(14,17)(19,22)(20,24)(27,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,6,8,10)(7,9,11,12)(13,15,18,23)(14,19,17,22)(16,25,26,21)(20,28,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,26)(14,20,17,24)(15,21,23,25)(19,27,22,28)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7,8,11)(6,9,10,12)(13,17,18,14)(15,22,23,19)(16,24,26,20)(21,28,25,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=fgh,d^2=fh,e^2=fg,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, 4851727);