#8843599

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

4154 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)(17,18,22,24)(19,20,27,23)(21,29,31,25)(26,32,30,28)(33,34,39,44)(35,42,37,40)(36,41,47,45)(38,46,48,43);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,19,22,27)(18,23,24,20)(21,30,31,26)(25,28,29,32)(35,37)(38,48)(40,42)(43,46);(2,3)(7,9)(8,10)(14,15)(18,24)(20,23)(25,29)(28,32)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,32)(28,29)(30,31)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,32)(24,29)(27,30)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=ghi,e^2=gh,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, 8750795);