#5593482

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

1518 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=h,e^2=hi,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, 5500678);