#5083812

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1614 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fg,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4991008);