#9390571

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

4038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=g,f^2=g,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=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297767);