#5522092

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

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fg,d^2=hi,e^2=fgh,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, 5429288);