#8119560

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	#55620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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)(3,9,4,6)(5,13,15,10)(11,14,12,16)(20,28)(24,29)(26,32)(30,31)(33,34,39,44)(35,37,42,40)(36,43,38,41)(45,48,46,47);(2,8)(3,4)(10,13)(11,12)(18,22)(23,25)(24,29)(30,31)(33,35)(34,40)(36,45)(37,44)(38,46)(39,42)(41,48)(43,47);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,37)(36,45)(38,46)(39,42)(40,44)(41,47)(43,48);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(33,36,39,38)(34,41,44,43)(35,45,42,46)(37,47,40,48);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,42)(35,44)(36,47)(38,48)(39,40)(41,46)(43,45);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,36)(34,43,44,41)(35,46,42,45)(37,48,40,47)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=i,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, 8026756);