#8551274

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458470);