#8695273

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8602469);