#8119250

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	#47591	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=ghi,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, 8026446);