#8418676

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	#42589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325872);