#8471183

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

4199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=i,e^2=hi,f^2=hi,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, 8378379);