#8661373

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8568569);