#8383788

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	#58238	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8290984);