#8696169

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

4129 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)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(34,36)(35,41)(40,44)(43,45);(1,3,8,7)(2,9,11,12)(4,14,5,13)(6,15,10,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,4,8,5)(2,10,11,6)(3,13,7,14)(9,15,12,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,5,8,4)(2,6,11,10)(3,14,7,13)(9,16,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,38,34)(35,42,41,39)(37,40,46,44)(43,47,45,48);(1,6)(2,4)(3,15)(5,11)(7,16)(8,10)(9,14)(12,13)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,7,8,3)(2,9,11,12)(4,14,5,13)(6,16,10,15)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37,38,46)(34,40,36,44)(35,43,41,45)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=g,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, 8603365);