#8425522

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332718);