#8551372

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

4274 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)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18)(19,23)(20,25)(21,27)(22,24)(26,28)(29,31)(30,32)(33,34,38,41)(35,42,37,39)(36,40,44,47)(43,48,45,46);(2,7)(3,12)(9,13)(10,16)(17,19)(18,23)(20,27)(21,25)(22,28)(24,26)(29,30)(31,32)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(2,7)(4,14)(8,11)(10,16)(18,24)(19,28)(25,32)(27,29)(36,44)(40,47)(43,45)(46,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,15)(14,16)(17,20)(18,25)(19,29)(21,26)(22,30)(23,31)(24,32)(27,28)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,11,15,8)(17,19)(18,26)(20,29)(21,25)(22,28)(23,24)(27,30)(31,32)(33,36,38,44)(34,40,41,47)(35,43,37,45)(39,46,42,48);(1,5)(2,10)(3,13)(4,11)(6,15)(7,16)(8,14)(9,12)(17,21)(18,27)(19,25)(20,23)(22,31)(24,29)(26,30)(28,32)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458568);