#5519639

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
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

1370 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)(11,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(2,5)(6,8)(9,10,13,16)(11,14,12,15)(18,22)(20,27)(23,29)(25,31)(37,45)(39,46)(42,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,32)(26,29)(30,31)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(9,11,13,12)(10,14,16,15)(17,19,21,26)(18,23,22,29)(20,25,27,31)(24,30,28,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20,21,27)(18,24,22,28)(19,25,26,31)(23,30,29,32)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5426835);