#8919327

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	#46777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(3,9)(4,5)(6,10)(7,13)(8,14)(11,16)(12,15)(33,34,38,39)(35,36,41,40)(37,45,47,42)(43,48,46,44);(1,3)(2,7)(4,11)(5,12)(6,13)(8,16)(9,10)(14,15)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(34,39)(36,40)(42,45)(44,48);(4,14)(5,8)(11,15)(12,16)(18,23)(19,26)(21,30)(27,29)(33,35,38,41)(34,40,39,36)(37,46,47,43)(42,44,45,48);(1,4)(2,8)(3,11)(5,10)(6,14)(7,16)(9,12)(13,15)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,48)(44,45)(46,47);(1,3)(2,9)(4,15)(5,12)(6,13)(7,10)(8,16)(11,14)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36,38,40)(34,41,39,35)(37,44,47,48)(42,46,45,43);(1,5,6,8)(2,4,10,14)(3,12,13,16)(7,11,9,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37)(34,42)(35,43)(36,44)(38,47)(39,45)(40,48)(41,46)

Polycyclic presentation:

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

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, 8826523);