#9040595

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	#48396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(9,11)(10,13)(12,14)(15,16)(33,34)(35,41)(36,47)(37,39)(38,42)(40,43)(44,45)(46,48);(2,3)(4,11)(8,10)(12,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,40,46)(34,41,43,48)(36,38,37,44)(39,45,47,42);(4,11)(5,13)(7,9)(8,10)(18,22)(20,29)(23,25)(26,30)(33,36,40,37)(34,39,43,47)(35,44,46,38)(41,42,48,45);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37,40,36)(34,39,43,47)(35,38,46,44)(41,42,48,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38)(34,42)(35,37)(36,46)(39,41)(40,44)(43,45)(47,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,39)(34,36)(35,45)(37,43)(38,41)(40,47)(42,46)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947791);