#8806743

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

4062 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)(3,5)(4,8)(6,10)(7,12)(9,14)(11,15)(13,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,3,6,12)(2,7,10,5)(4,11,14,16)(8,13,9,15)(18,23)(20,25)(24,29)(28,31)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,48)(45,46);(1,4,6,14)(2,8,10,9)(3,11,12,16)(5,15,7,13)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34,39,38)(35,40,37,42)(36,44,46,41)(43,48,45,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,15)(8,10)(12,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,5)(2,3)(4,13)(6,7)(8,16)(9,11)(10,12)(14,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37,39,35)(34,40,38,42)(36,45,46,43)(41,47,44,48);(1,3)(2,5)(4,11)(6,12)(7,10)(8,15)(9,13)(14,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,38,39,34)(35,40,37,42)(36,44,46,41)(43,47,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8713939);