#8417481

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	#44975	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3919 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,24,23)(19,27,26,22)(20,30,21,25)(28,31,29,32)(34,38)(36,43)(39,44)(42,47);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(17,19,24,26)(18,22,23,27)(20,28,21,29)(25,31,30,32)(35,41)(39,44)(42,47)(46,48);(3,12)(6,13)(8,16)(10,15)(17,20,24,21)(18,25,23,30)(19,28,26,29)(22,31,27,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,21,24,20)(18,25,23,30)(19,29,26,28)(22,31,27,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,22,24,27)(18,26,23,19)(20,31,21,32)(25,29,30,28)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)(17,23,24,18)(19,27,26,22)(20,25,21,30)(28,31,29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,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, 8324677);