#8720634

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(3,9)(4,7)(5,10)(6,12)(8,13)(11,15)(14,16)(17,18,23,27)(19,20,30,24)(21,31,29,25)(22,26,28,32)(33,34)(35,36)(37,38)(39,41)(40,43)(42,48)(44,46)(45,47);(1,3)(2,6)(4,11)(5,12)(7,14)(8,15)(9,10)(13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,32)(26,31)(33,35,40,41)(34,36,43,39)(37,47,42,44)(38,45,48,46);(1,4,5,13)(2,7,10,8)(3,11,12,16)(6,14,9,15)(17,20,23,24)(18,19,27,30)(21,26,29,32)(22,25,28,31)(33,36,40,39)(34,41,43,35)(37,45,42,46)(38,44,48,47);(1,4)(2,8)(3,11)(5,13)(6,15)(7,10)(9,14)(12,16)(17,20)(18,19)(21,26)(22,31)(23,24)(25,28)(27,30)(29,32)(33,37,40,42)(34,38,43,48)(35,44,41,47)(36,46,39,45);(1,3)(2,9)(4,11)(5,12)(6,10)(7,15)(8,14)(13,16)(17,21)(18,25)(19,28)(20,26)(22,30)(23,29)(24,32)(27,31)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46);(1,2)(3,6)(4,7)(5,10)(8,13)(9,12)(11,14)(15,16)(17,22,23,28)(18,26,27,32)(19,29,30,21)(20,25,24,31)(33,39,40,36)(34,41,43,35)(37,46,42,45)(38,44,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=i,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, 8627830);