#8197108

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=ghi,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, 8104304);