#8755113

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	#56523	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(18,22)(19,27)(20,30)(26,32)(33,34)(35,40)(36,45)(37,38)(39,42)(41,48)(43,47)(44,46);(1,3,7,5)(2,8,4,9)(6,13,14,11)(10,16,12,15)(17,18,23,22)(19,28,27,24)(20,25,30,31)(21,32,29,26)(33,35,39,36)(34,40,42,45)(37,46,41,43)(38,44,48,47);(2,4)(8,9)(10,12)(15,16)(17,19,23,27)(18,24,22,28)(20,29,30,21)(25,32,31,26)(33,36,39,35)(34,40,42,45)(37,46,41,43)(38,47,48,44);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,21,23,29)(18,26,22,32)(19,20,27,30)(24,25,28,31)(33,37,39,41)(34,38,42,48)(35,43,36,46)(40,47,45,44);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,22,23,18)(19,28,27,24)(20,31,30,25)(21,32,29,26)(33,38)(34,41)(35,44)(36,47)(37,42)(39,48)(40,43)(45,46)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662309);