#8118830

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026026);