#8467144

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

4180 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)(3,8,5,4)(6,9,15,14)(11,16,13,12)(17,18,23,22)(19,24,29,27)(20,25,30,32)(21,31,28,26)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3,7,5)(2,4,10,8)(6,11,15,13)(9,12,14,16)(17,18,23,22)(19,27,29,24)(20,25,30,32)(21,26,28,31)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,4,7,8)(2,3,10,5)(6,12,15,16)(9,11,14,13)(17,19,23,29)(18,24,22,27)(20,28,30,21)(25,26,32,31)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,2)(3,4)(5,8)(6,14)(7,10)(9,15)(11,16)(12,13)(17,20)(18,25)(19,28)(21,29)(22,32)(23,30)(24,26)(27,31)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,5,7,3)(2,8,10,4)(6,13,15,11)(9,16,14,12)(17,21,23,28)(18,26,22,31)(19,20,29,30)(24,32,27,25)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,16)(10,14)(17,22,23,18)(19,27,29,24)(20,25,30,32)(21,31,28,26)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=h,e^2=ghi,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, 8374340);