#8720458

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627654);