#8285316

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	#54850	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8192512);