#8712137

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

4007 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,38,42)(35,39,40,36)(37,41,47,48)(43,45,46,44);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,45)(44,48)(46,47);(2,9)(4,15)(8,11)(12,14)(19,28)(20,29)(24,30)(25,32)(33,36,38,39)(34,40,42,35)(37,44,47,45)(41,46,48,43);(1,4,7,15)(2,10,9,6)(3,12,5,14)(8,13,11,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36)(34,35)(37,45)(38,39)(40,42)(41,46)(43,48)(44,47);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,13,10,16)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,35)(34,36)(37,46)(38,40)(39,42)(41,45)(43,47)(44,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,37)(34,41)(35,43)(36,44)(38,47)(39,45)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8619333);