#8464211

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	#45838	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=gi,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, 8371407);