#9712823

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620019);