#8417030

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	#52873	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,9,13,16)(4,10,12,6)(5,11,14,15)(17,18,24,23)(19,27,22,25)(20,30,26,21)(28,29,31,32)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(2,8)(4,12)(5,14)(9,16)(17,19,24,22)(18,25,23,27)(20,31,26,28)(21,32,30,29)(34,40)(35,42)(41,44)(43,48);(1,3,7,13)(2,9,8,16)(4,5,12,14)(6,15,10,11)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,29)(27,32)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4)(2,10)(3,5)(6,8)(7,12)(9,11)(13,14)(15,16)(17,21,24,30)(18,26,23,20)(19,29,22,32)(25,31,27,28)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5)(2,11)(3,12)(4,13)(6,9)(7,14)(8,15)(10,16)(17,22,24,19)(18,27,23,25)(20,31,26,28)(21,32,30,29)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6,7,10)(2,4,8,12)(3,11,13,15)(5,9,14,16)(17,23,24,18)(19,27,22,25)(20,30,26,21)(28,32,31,29)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=i,f^2=g,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, 8324226);