#8493227

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

4182 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8400423);