#8919226

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

4049 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)(3,4,8,12)(5,13,14,9)(10,16,15,11)(17,18,23,27)(19,28,21,24)(20,22,26,25)(29,32,31,30);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32)(33,34,38,39)(35,40,42,47)(36,37,41,46)(43,48,45,44);(2,7)(3,8)(5,14)(11,16)(20,26)(22,25)(29,31)(30,32)(34,39)(35,42)(37,46)(43,45);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30)(33,35)(34,40)(36,45)(37,44)(38,42)(39,47)(41,43)(46,48);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,21,23,19)(18,24,27,28)(20,29,26,31)(22,32,25,30)(33,36,38,41)(34,37,39,46)(35,43,42,45)(40,48,47,44);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)(33,37)(34,41)(35,44)(36,39)(38,46)(40,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826422);