#9203612

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	#55050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,24)(20,28)(21,23)(22,27)(25,30)(26,31)(29,32)(33,34,38,39)(35,41,42,36)(37,45,47,40)(43,44,48,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,21,27)(18,22,23,24)(20,26,30,29)(25,31,28,32)(33,34,38,39)(35,36,42,41)(37,40,47,45)(43,44,48,46);(3,10)(7,8)(11,13)(12,16)(18,23)(22,24)(25,28)(31,32)(34,39)(35,42)(40,45)(43,48);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,18)(19,22)(20,25)(21,23)(24,27)(26,31)(28,30)(29,32)(33,35,38,42)(34,36,39,41)(37,43,47,48)(40,44,45,46);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,19)(18,24)(20,29)(21,27)(22,23)(25,31)(26,30)(28,32)(33,36)(34,35)(37,46)(38,41)(39,42)(40,48)(43,45)(44,47);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,20)(18,25)(19,26)(21,30)(22,31)(23,28)(24,32)(27,29)(33,37,38,47)(34,40,39,45)(35,43,42,48)(36,44,41,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=ghi,f^2=gh,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, 9110808);