#4747277

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1460 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=fg,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4654473);