#8761956

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

4270 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669152);