#8855128

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

4196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8762324);