#4908010

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

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fh,d^2=fgh,e^2=i,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, 4815206);