#8930312

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

4293 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(19,25)(23,30)(26,28)(31,32)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(1,2,5,8)(3,6,9,10)(4,7,12,14)(11,13,15,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(3,9)(4,12)(6,10)(7,14)(18,22)(19,25)(24,27)(26,28)(34,36)(38,42)(40,44)(45,48);(1,3)(2,6)(4,11)(5,9)(7,13)(8,10)(12,15)(14,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,36,39,34)(35,42,41,38)(37,40,46,44)(43,45,47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,4,5,12)(2,7,8,14)(3,11,9,15)(6,13,10,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=hi,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, 8837508);