#8456976

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

4123 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)(10,13)(11,12)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28)(33,34,39,43)(35,40,46,44)(36,38,42,48)(37,47,45,41);(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35,39,46)(34,40,43,44)(36,45,42,37)(38,41,48,47);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,36)(34,38)(35,45)(37,46)(39,42)(40,41)(43,48)(44,47);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=h,e^2=ghi,f^2=g,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, 8364172);