#8843497

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	#51142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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)(19,24)(20,30)(21,26)(22,23)(25,29)(27,32)(28,31)(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)(18,23)(21,30)(24,25)(28,32)(33,34,39,42)(35,40,37,36)(38,46,47,41)(43,44,45,48);(2,3)(7,9)(8,10)(14,15)(17,19,22,29)(18,24,23,25)(20,27,26,31)(21,28,30,32)(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)(17,19)(18,25)(20,31)(21,28)(22,29)(23,24)(26,27)(30,32)(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)(17,20,22,26)(18,21,23,30)(19,27,29,31)(24,28,25,32)(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)(17,21)(18,26)(19,28)(20,23)(22,30)(24,31)(25,27)(29,32)(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=i,c^2=gh,d^2=hi,e^2=gh,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, 8750693);