#1747072

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

1710 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)(7,10,12,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,6,9,8)(7,11,12,10)(14,18)(16,24)(19,21)(22,28)(32,41)(35,40)(38,44)(42,43);(5,7,9,12)(6,10,8,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(1,2,3,4)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43);(5,8,9,6)(7,11,12,10)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=i,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, 1654268);