#8806778

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

3912 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)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,18,23,20)(19,26,25,22)(21,24,30,28)(27,32,31,29)(34,40)(35,43)(36,46)(38,48);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,18,23,20)(19,22,25,26)(21,28,30,24)(27,32,31,29)(33,34,39,40)(35,41,43,37)(36,42,46,45)(38,47,48,44);(1,2,5,8)(3,9,11,6)(4,7,14,12)(10,16,13,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,41)(36,47)(37,40)(38,45)(39,43)(42,48)(44,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36)(34,42)(35,44)(37,48)(38,41)(39,46)(40,45)(43,47);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,37,39,41)(34,43,40,35)(36,48,46,38)(42,47,45,44);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,48)(34,44,40,47)(35,45,43,42)(36,41,46,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8713974);