#8843510

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	#51519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=ghi,e^2=h,f^2=gh,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, 8750706);