#8761943

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	#47994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^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, 8669139);