#8430472

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8337668);