#8930245

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 8837441);