#8661371

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	#46102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 8568567);