#8762092

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

4274 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)(5,13,15,8)(10,12,14,16)(17,18,22,23)(19,24,29,25)(20,26,31,21)(27,28,30,32)(33,34,38,41)(35,37,44,47)(36,40,46,39)(42,48,43,45);(1,3,6,11)(2,4,7,9)(5,14,15,10)(8,16,13,12)(18,23)(21,26)(24,25)(28,32)(33,35,38,44)(34,37,41,47)(36,43,46,42)(39,48,40,45);(2,7)(4,9)(8,13)(12,16)(17,19,22,29)(18,24,23,25)(20,30,31,27)(21,28,26,32)(33,36,38,46)(34,39,41,40)(35,42,44,43)(37,45,47,48);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,24)(26,32)(30,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,14)(15,16)(17,20)(18,26)(19,27)(21,23)(22,31)(24,28)(25,32)(29,30)(33,35)(34,37)(36,43)(38,44)(39,48)(40,45)(41,47)(42,46);(1,5,6,15)(2,8,7,13)(3,10,11,14)(4,12,9,16)(17,21,22,26)(18,20,23,31)(19,28,29,32)(24,30,25,27)(33,37)(34,35)(36,45)(38,47)(39,42)(40,43)(41,44)(46,48)

Polycyclic presentation:

2^9|a^2=gi,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, 8669288);