#8339947

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	#56192	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18,24,28)(19,27,22,23)(20,26,21,25)(29,32,30,31)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19,24,22)(18,23,28,27)(20,29,21,30)(25,32,26,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3,6,4)(2,8,10,7)(5,11,13,12)(9,15,16,14)(17,20,24,21)(18,25,28,26)(19,29,22,30)(23,32,27,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,21,24,20)(18,26,28,25)(19,29,22,30)(23,32,27,31)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,22,24,19)(18,27,28,23)(20,30,21,29)(25,31,26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,23,24,27)(18,19,28,22)(20,31,21,32)(25,30,26,29)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8247143);