#4942505

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

1772 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)(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(30,31)(33,41)(35,37)(39,44);(14,18)(16,24)(19,21)(22,28)(29,30,34,31)(32,35,40,37)(33,38,41,36)(39,43,44,42);(6,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,31,34,30)(32,37,40,35)(33,38,41,36)(39,43,44,42);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,33)(30,36)(31,38)(32,39)(34,41)(35,42)(37,43)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fh,d^2=f,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, 4849701);