#8188523

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,23,24)(19,28,29,25)(20,32,26,21)(22,27,30,31)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(2,3)(7,9)(8,10)(14,15)(18,24)(19,29)(20,26)(27,31)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(5,13)(8,10)(11,16)(14,15)(17,19)(18,25)(20,30)(21,31)(22,26)(23,29)(24,28)(27,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,26)(18,21,24,32)(19,30,29,22)(25,31,28,27)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,26)(19,31)(20,24)(22,25)(23,32)(27,29)(28,30)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,30)(18,27,24,31)(19,26,29,20)(21,25,32,28)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=ghi,e^2=gh,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, 8095719);