#8926534

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	#46488	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833730);