#8841348

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748544);