#4381348

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1722 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,21,22,19)(16,24,26,20)(23,27,25,28)(33,41)(36,40)(38,44)(42,43);(14,18)(15,22)(20,24)(23,25)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,7,8,11)(6,9,10,12)(13,16)(14,20)(15,23)(17,26)(18,24)(19,27)(21,28)(22,25)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

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, 4288544);