#8992975

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	#54952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8900171);