#8724853

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8632049);