#8843378

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

4211 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,22,23)(19,25,26,24)(20,29,31,21)(27,32,30,28)(36,43)(39,41)(44,48)(46,47);(2,7)(4,13)(5,8)(11,14)(18,23)(21,29)(24,25)(28,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,7)(3,10)(9,12)(11,14)(19,26)(24,25)(27,30)(28,32)(34,35)(39,41)(40,42)(46,47);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,19)(18,24)(20,30)(21,28)(22,26)(23,25)(27,31)(29,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20)(18,21)(19,27)(22,31)(23,29)(24,32)(25,28)(26,30)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,21)(18,20)(19,28)(22,29)(23,31)(24,30)(25,27)(26,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=hi,e^2=h,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, 8750574);