#8464272

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=i,f^2=gi,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, 8371468);