#1680000

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fg,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1587196);