#9136915

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	#43216	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9044111);