#8406479

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	#43442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313675);