#8904776

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	#47992	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

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,d^c=dgi,f^c=fgi,e^d=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, 8811972);