#8406481

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,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, 8313677);