#8720823

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,34,39,42)(35,40,37,41)(36,45,47,38)(43,44,46,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,30)(28,29)(31,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,48)(44,46)(45,47);(2,9)(4,15)(8,11)(12,14)(18,23)(19,24)(25,32)(27,29)(33,35,39,37)(34,40,42,41)(36,46,47,43)(38,44,45,48);(1,4,7,15)(2,10,9,6)(3,12,5,14)(8,13,11,16)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28)(33,36,39,47)(34,38,42,45)(35,43,37,46)(40,48,41,44);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,13,10,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,37,39,35)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)(33,38)(34,36)(35,44)(37,48)(39,45)(40,46)(41,43)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8628019);