#6252672

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1508 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fghi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6159868);