#3855784

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1368 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=hi,d^2=fghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3762980);