#8095534

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002730);