#4762801

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1459 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)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,25)(21,24)(23,27)(26,29)(28,31)(30,32)(34,36)(39,41)(40,44)(47,48);(9,10)(11,13)(12,14)(15,16)(18,19)(20,27)(24,26)(28,32)(33,34,38,36)(35,41,43,39)(37,40,46,44)(42,48,45,47);(11,16)(13,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(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)(10,13)(12,16)(14,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=h,d^2=gh,e^2=fghi,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, 4669997);