#8734710

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

4303 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=g,e^2=h,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, 8641906);