#8334327

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	#55291	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,5,10,8)(4,12,15,9)(11,16,13,14)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,39,36)(35,38,41,42)(37,40,46,44)(43,45,47,48);(2,7)(3,10)(9,12)(11,13)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(2,7)(5,8)(9,12)(14,16)(18,23)(19,25)(24,32)(26,27)(35,41)(38,42)(43,47)(45,48);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,36,39,34)(35,42,41,38)(37,40,46,44)(43,45,47,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,19)(18,21)(20,26)(22,25)(23,31)(24,30)(27,29)(28,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^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, 8241523);