#8920757

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

4057 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,e^2=i,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, 8827953);