#8930182

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

4135 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)(3,8,11,15)(4,9,13,12)(5,10,16,14)(19,25)(23,30)(26,28)(31,32)(33,34,40,43)(35,44,37,41)(36,39,38,42)(45,48,46,47);(2,7)(4,13)(8,15)(10,14)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,40,37)(34,41,43,44)(36,45,38,46)(39,47,42,48);(3,11)(5,16)(8,15)(10,14)(18,22)(19,25)(24,27)(26,28)(33,36,40,38)(34,42,43,39)(35,45,37,46)(41,48,44,47);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,37,40,35)(34,41,43,44)(36,45,38,46)(39,48,42,47);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,38,40,36)(34,42,43,39)(35,46,37,45)(41,48,44,47);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,39,40,42)(34,36,43,38)(35,47,37,48)(41,45,44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837378);