#8110462

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	#53298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017658);