#8664420

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571616);