#1982570

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=ghi,d^2=ghi,e^2=fhi,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, 1889766);