#8772910

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

4092 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)(3,8,13,9)(4,10,15,5)(11,12,14,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,37)(35,38,42,40)(36,44,46,41)(43,47,48,45);(2,7)(5,10)(8,9)(12,16)(21,29)(25,28)(27,30)(31,32)(34,37)(35,42)(41,44)(43,48);(1,3,6,13)(2,8,7,9)(4,14,15,11)(5,12,10,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,39,42)(34,40,37,38)(36,43,46,48)(41,47,44,45);(1,3)(2,9)(4,11)(5,12)(6,13)(7,8)(10,16)(14,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36,39,46)(34,41,37,44)(35,43,42,48)(38,45,40,47);(1,4)(2,10)(3,11)(5,7)(6,15)(8,12)(9,16)(13,14)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,37,39,34)(35,38,42,40)(36,44,46,41)(43,45,48,47);(1,5,6,10)(2,4,7,15)(3,12,13,16)(8,14,9,11)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=h,e^2=h,f^2=gh,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, 8680106);