#8119299

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	#47700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026495);