#8186848

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	#42570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(34,36)(35,41)(40,44)(43,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(2,8)(3,9)(10,13)(11,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

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

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, 8094044);