#8492946

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

3956 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)(4,12)(5,14)(6,11)(7,8)(10,15)(13,16)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(2,8)(5,15)(9,12)(11,13)(19,28)(21,29)(24,32)(25,30)(34,39)(35,42)(41,44)(43,48);(1,3,7,4)(2,9,8,12)(5,11,15,13)(6,10,16,14)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,7,3)(2,9,8,12)(5,13,15,11)(6,10,16,14)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,15)(2,10,8,14)(3,11,4,13)(6,9,16,12)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,6,7,16)(2,11,8,13)(3,10,4,14)(5,9,15,12)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=i,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, 8400142);