#8762090

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

4098 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)(3,9,11,4)(5,13,14,8)(10,12,16,15)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(2,7)(4,9)(5,14)(10,16)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(34,39)(35,42)(41,44)(43,48);(1,3,6,11)(2,4,7,9)(5,10,14,16)(8,12,13,15)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669286);