#8322116

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

4021 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:

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

Polycyclic presentation:

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

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, 8229312);