#8928675

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
prime radical	#46447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34)(35,39)(36,40)(37,45)(38,42)(41,47)(43,44)(46,48);(1,2,5,8)(3,6,9,10)(4,7,12,14)(11,13,15,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(33,35)(34,39)(36,42)(37,43)(38,40)(41,48)(44,45)(46,47);(3,9)(4,12)(6,10)(7,14)(19,27)(20,28)(22,30)(24,32)(33,35,38,40)(34,36,42,39)(37,46,47,43)(41,48,45,44);(1,3,5,9)(2,6,8,10)(4,11,12,15)(7,13,14,16)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,4)(2,7)(3,11)(5,12)(6,13)(8,14)(9,15)(10,16)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45)

Polycyclic presentation:

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

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, 8835871);