#8924985

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	#54923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832181);