#8107681

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	#40673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014877);