#8402447

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309643);