#8720699

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

4097 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,25)(19,27,22,24)(20,30,26,21)(28,32,31,29)(33,34)(35,37)(36,45)(38,40)(39,42)(41,44)(43,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,22)(18,24,25,27)(20,31,26,28)(21,29,30,32)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(4,13)(7,12)(10,16)(14,15)(18,25)(20,26)(24,27)(28,31)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20,23,26)(18,21,25,30)(19,28,22,31)(24,32,27,29)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21)(18,26)(19,29)(20,25)(22,32)(23,30)(24,28)(27,31)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,22,23,19)(18,24,25,27)(20,28,26,31)(21,29,30,32)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627895);