#6034434

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

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, 5941630);