#9039178

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

4115 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(2,7)(4,13)(8,15)(10,14)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(4,13)(5,16)(9,11)(10,14)(18,24)(21,30)(23,26)(27,28)(34,36)(37,45)(39,41)(42,48);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)(17,20,22,19)(18,23,24,26)(21,28,30,27)(25,31,29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4,6,13)(2,9,7,11)(3,10,12,14)(5,15,16,8)(17,18)(19,23)(20,26)(21,25)(22,24)(27,31)(28,32)(29,30)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,5)(2,10)(3,11)(4,15)(6,16)(7,14)(8,13)(9,12)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946374);