#8209996

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8117192);