#8165363

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

3942 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8072559);