#9059108

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

4152 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,16)(14,15)(17,18)(19,27)(20,25)(21,29)(22,23)(24,28)(26,31)(30,32)(33,34)(35,40)(36,42)(37,43)(38,46)(39,41)(44,48)(45,47);(2,7)(3,12)(4,10)(14,16)(18,23)(19,28)(25,32)(26,29)(33,35)(34,40)(36,39)(37,44)(38,45)(41,42)(43,48)(46,47);(3,12)(4,10)(5,9)(8,11)(18,23)(19,28)(20,30)(21,31)(34,41)(35,36)(37,47)(38,48);(1,3,6,12)(2,8,7,11)(4,15,10,13)(5,14,9,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,30)(31,32)(33,36,39,35)(34,42,41,40)(37,44,47,46)(38,43,48,45);(1,4,6,10)(2,9,7,5)(3,13,12,15)(8,14,11,16)(17,20)(18,25)(19,29)(21,27)(22,30)(23,32)(24,31)(26,28)(33,37,39,47)(34,43,41,45)(35,44,36,46)(38,40,48,42);(1,5,6,9)(2,10,7,4)(3,14,12,16)(8,13,11,15)(17,21,22,31)(18,26,23,29)(19,25,28,32)(20,27,30,24)(33,38)(34,44)(35,45)(36,43)(37,40)(39,48)(41,46)(42,47)

Polycyclic presentation:

2^9|d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8966304);