#1672337

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

1708 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)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30)(31,36)(32,39)(33,35)(34,41)(37,42)(38,44)(40,43);(7,11)(10,12)(14,18)(16,24)(19,21)(22,28)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,7)(6,10)(8,12)(9,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(5,8,9,6)(7,12,11,10)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(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=g,c^2=g,d^2=fg,e^2=h,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, 1579533);