#8445867

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(19,29)(20,30)(24,28)(25,32)(34,36)(39,41)(40,44)(46,47);(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,e^2=i,f^2=g,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, 8353063);