#8107680

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	#52916	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 8014876);