#5486341

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 4194304
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1630 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)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,45)(46,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(20,28)(23,30)(26,32)(29,31)(34,39)(35,40)(41,47)(43,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35)(34,36)(37,45)(38,40)(39,42)(41,48)(43,46)(44,47);(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36)(34,40)(35,39)(37,44)(38,42)(41,45)(43,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37,38,46)(34,41,39,47)(35,43,40,45)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=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, 5393537);