#8211614

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	#56739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

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

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, 8118810);