#8271335

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

3861 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,38)(39,44)(40,45)(46,48);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(18,19)(23,25)(24,26)(30,31)(35,41)(39,44)(42,47)(46,48);(2,7)(6,9)(8,12)(15,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

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

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, 8178531);