#5451711

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1298 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5358907);