#8504740

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

3966 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8411936);