#8772791

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

4056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gh,e^2=gh,f^2=h,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, 8679987);