#4871229

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1504 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)(11,14)(12,16)(13,15)(20,27)(24,29)(25,31)(30,32)(33,34,38,35)(36,39,44,41)(37,42,46,40)(43,48,45,47);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(18,22)(23,28)(24,29)(30,32)(34,35)(39,41)(40,42)(47,48);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,34)(36,41,44,39)(37,42,46,40)(43,48,45,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,44)(34,39,35,41)(37,45,46,43)(40,48,42,47);(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,46)(34,40,35,42)(36,43,44,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4778425);