#4456689

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1374 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=fi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4363885);