#3096626

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

1306 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fgh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3003822);