#8227355

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

3848 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:

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

Polycyclic presentation:

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

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, 8134551);