#8691833

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

4208 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)(3,7,10,14)(4,8,13,11)(9,15,12,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(34,36)(35,41)(40,44)(43,45);(2,6)(7,14)(8,11)(15,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36,38,34)(35,42,41,39)(37,40,46,44)(43,47,45,48);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(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=h,e^2=g,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, 8599029);