#8193752

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

4165 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(5,13)(8,10)(11,16)(14,15)(18,22)(23,25)(24,29)(30,31)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8100948);