#8269733

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	#54863	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8176929);