#8463491

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

3952 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8370687);