#8755688

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

3977 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662884);