#8922851

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18)(19,25)(20,29)(21,26)(22,27)(23,28)(24,30)(31,32)(33,34,39,41)(35,40,45,48)(36,42,46,47)(37,43,44,38);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19)(18,23)(20,26)(21,24)(22,28)(25,27)(29,32)(30,31)(33,35,39,45)(34,40,41,48)(36,44,46,37)(38,47,43,42);(2,4)(5,13)(7,8)(10,16)(20,30)(21,32)(24,29)(26,31)(34,41)(35,45)(36,46)(38,43);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20)(18,24)(19,26)(21,23)(22,30)(25,32)(27,29)(28,31)(33,36)(34,42)(35,44)(37,45)(38,40)(39,46)(41,47)(43,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,24)(22,28)(23,27)(26,30)(29,32)(33,37,39,44)(34,43,41,38)(35,46,45,36)(40,47,48,42);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,21)(18,26)(19,24)(20,23)(22,32)(25,30)(27,31)(28,29)(33,38,39,43)(34,44,41,37)(35,47,45,42)(36,40,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=i,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, 8830047);