#8751860

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	#46523	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 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)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,39,44)(35,42,37,40)(36,43,38,41)(45,47,46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(2,6)(7,10)(8,12)(14,16)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,42,44,40)(36,45,38,46)(41,47,43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,38,39,36)(34,43,44,41)(35,45,37,46)(40,47,42,48)

Polycyclic presentation:

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

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, 8659056);