#8492948

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

4106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=gi,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, 8400144);