#8745973

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653169);