#8734504

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

4113 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)(3,11,5,8)(4,10,15,6)(12,16,14,13)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34)(35,44)(36,41)(37,40)(38,42)(39,43)(45,46)(47,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35,39,37)(34,40,43,44)(36,45,42,47)(38,46,41,48);(2,9)(4,15)(8,11)(12,14)(19,27)(20,28)(23,32)(24,29)(33,36,39,42)(34,41,43,38)(35,45,37,47)(40,48,44,46);(1,4,7,15)(2,10,9,6)(3,12,5,14)(8,13,11,16)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36)(34,41)(35,45)(37,47)(38,43)(39,42)(40,48)(44,46);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,13,10,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,37,39,35)(34,40,43,44)(36,47,42,45)(38,46,41,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,38,39,41)(34,42,43,36)(35,46,37,48)(40,47,44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641700);