#8712455

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	#54839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,7)(4,13)(5,10)(6,8)(9,14)(11,15)(12,16)(17,18)(19,27)(20,29)(21,24)(22,28)(23,25)(26,31)(30,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,6,11)(2,7,8,15)(4,12,14,10)(5,13,16,9)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,26,32,29)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(2,8)(3,11)(9,13)(10,12)(18,25)(20,31)(24,27)(28,30)(35,44)(36,45)(40,46)(41,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,15)(10,14)(13,16)(17,20)(18,26)(19,28)(21,30)(22,24)(23,31)(25,29)(27,32)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,4,6,14)(2,9,8,13)(3,12,11,10)(5,15,16,7)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,26,32,29)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,5)(2,10)(3,13)(4,7)(6,16)(8,12)(9,11)(14,15)(17,22,23,32)(18,28,25,30)(19,26,21,29)(20,27,31,24)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,e^2=h,f^2=g,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, 8619651);