#8922788

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

4209 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8829984);