#8893510

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	#40044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800706);