#8534918

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442114);