#1966378

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

1366 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)(11,14,16,15)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,38,39)(35,36,40,44)(37,41,45,46)(42,43,47,48);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,15,16,14)(19,26)(22,25)(27,32)(30,31)(34,39)(35,40)(37,45)(43,48);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,47)(46,48);(1,3,5,8)(2,6,4,7)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37)(34,41)(35,42)(36,43)(38,45)(39,46)(40,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fghi,d^2=gi,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, 1873574);