#8845529

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

4237 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32)(33,34,39,42)(35,40,36,41)(37,47,48,38)(43,44,45,46);(2,7)(4,8)(5,14)(11,16)(18,23)(21,25)(24,26)(29,32)(33,35,39,36)(34,40,42,41)(37,43,48,45)(38,46,47,44);(2,7)(3,9)(10,13)(11,16)(19,27)(24,26)(28,30)(29,32)(35,36)(40,41)(43,45)(44,46);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,27)(18,24,23,26)(20,30,31,28)(21,29,25,32)(33,36,39,35)(34,41,42,40)(37,43,48,45)(38,46,47,44);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,37)(34,38)(35,43)(36,45)(39,48)(40,46)(41,44)(42,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,25)(18,20,23,31)(19,29,27,32)(24,30,26,28)(33,38)(34,37)(35,44)(36,46)(39,47)(40,45)(41,43)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 8752725);