#8464318

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	#53522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=hi,f^2=i,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, 8371514);