#9206633

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	#41468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,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, 9113829);