#8417209

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	#52743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(2,6)(3,9)(8,13)(11,15)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(4,12)(8,13)(11,15)(14,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35,38,46)(34,39,42,43)(36,44,37,45)(40,47,41,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324405);