#8762091

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

4313 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)(3,12)(4,13)(5,11)(6,16)(7,9)(8,14)(10,15)(17,18)(19,21)(20,25)(22,23)(24,27)(26,29)(28,32)(30,31)(33,34,38,41)(35,37,44,47)(36,40,46,39)(42,48,43,45);(1,3,7,14)(2,8,9,12)(4,5,15,16)(6,10,11,13)(18,23)(21,24)(25,31)(29,32)(33,35,38,44)(34,37,41,47)(36,43,46,42)(39,48,40,45);(2,9)(4,15)(5,16)(8,12)(17,19,22,27)(18,21,23,24)(20,28,30,26)(25,32,31,29)(33,36,38,46)(34,39,41,40)(35,42,44,43)(37,45,47,48);(1,4)(2,10)(3,5)(6,12)(7,15)(8,11)(9,13)(14,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,5)(2,11)(3,4)(6,9)(7,16)(8,10)(12,13)(14,15)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28)(33,35)(34,37)(36,43)(38,44)(39,48)(40,45)(41,47)(42,46);(1,6,7,11)(2,5,9,16)(3,13,14,10)(4,12,15,8)(17,21,22,24)(18,19,23,27)(20,29,30,32)(25,26,31,28)(33,37)(34,35)(36,45)(38,47)(39,42)(40,43)(41,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,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, 8669287);