#9203624

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	#52957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,23,25)(19,28,22,24)(20,26,31,30)(21,27,32,29)(34,39)(35,41)(37,45)(44,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,26,32,30)(34,39)(36,42)(40,47)(44,46);(3,10)(7,8)(11,13)(12,16)(18,25)(20,31)(24,28)(27,29)(33,34,38,39)(35,42,41,36)(37,40,45,47)(43,46,48,44);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,31)(24,32)(25,30)(33,35,38,41)(34,36,39,42)(37,43,45,48)(40,44,47,46);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,24,31,28)(33,36)(34,35)(37,46)(38,42)(39,41)(40,48)(43,47)(44,45);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,30,32,26)(33,37,38,45)(34,40,39,47)(35,43,41,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110820);