#8472874

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

4079 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,43)(36,42)(37,47)(38,40)(39,41)(44,45)(46,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35,40,39)(34,41,38,43)(36,44,47,46)(37,45,42,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36,40,47)(34,42,38,37)(35,44,39,46)(41,48,43,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,37)(34,36)(35,45)(38,47)(39,48)(40,42)(41,44)(43,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,38,40,34)(35,41,39,43)(36,37,47,42)(44,48,46,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,39,40,35)(34,41,38,43)(36,46,47,44)(37,45,42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8380070);