#8188347

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

3963 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)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,39,41)(35,42,36,38)(37,45,47,40)(43,46,44,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,39,41)(35,42,36,38)(37,40,47,45)(43,48,44,46);(5,13)(8,10)(11,16)(14,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,35,39,36)(34,38,41,42)(37,43,47,44)(40,46,45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,36,39,35)(34,38,41,42)(37,44,47,43)(40,46,45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37,39,47)(34,40,41,45)(35,43,36,44)(38,46,42,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,38)(34,35)(36,41)(37,46)(39,42)(40,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gi,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, 8095543);