#1735973

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

1356 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=h,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, 1643169);