#1576872

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

1789 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fgi,d^2=g,e^2=fg,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, 1484068);