#8664392

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	#56306	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8571588);