#4524169

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1423 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4431365);