#8457061

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	#53610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(18,24)(19,20)(26,31)(27,28)(33,34,39,44)(35,40,46,48)(36,38,43,41)(37,42,45,47);(1,3)(2,8)(4,6)(5,13)(7,14)(9,10)(11,12)(15,16)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,29)(30,31)(33,35,39,46)(34,40,44,48)(36,37,43,45)(38,42,41,47);(1,4,7,5)(2,9,11,15)(3,6,14,13)(8,10,12,16)(17,19,23,20)(18,25,24,22)(21,27,30,28)(26,32,31,29)(33,36,39,43)(34,41,44,38)(35,45,46,37)(40,42,48,47);(1,5,7,4)(2,9,11,15)(3,13,14,6)(8,10,12,16)(17,20,23,19)(18,25,24,22)(21,28,30,27)(26,32,31,29)(33,35)(34,40)(36,45)(37,43)(38,47)(39,46)(41,42)(44,48);(1,6,7,13)(2,10,11,16)(3,4,14,5)(8,9,12,15)(17,21,23,30)(18,26,24,31)(19,27,20,28)(22,29,25,32)(33,37,39,45)(34,42,44,47)(35,43,46,36)(38,40,41,48);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22,23,25)(18,20,24,19)(21,29,30,32)(26,28,31,27)(33,38,39,41)(34,43,44,36)(35,42,46,47)(37,40,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=g,e^2=gi,f^2=h,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, 8364257);