#8909777

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,f^2=h,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, 8816973);