#1965932

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,15,16,14)(18,19)(20,27)(21,29)(30,31)(36,45)(39,44)(42,48)(46,47);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,16,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fhi,d^2=ghi,e^2=fghi,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, 1873128);