#8196908

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(18,23)(19,26)(20,28)(29,31)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(36,44)(40,46)(43,47)(45,48);(2,6)(7,13)(8,11)(14,15)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=ghi,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, 8104104);