#8648914

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556110);