#8810340

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

4241 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717536);