#9924012

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,37,40)(35,38,43,41)(36,39,45,48)(42,46,44,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(19,27)(21,30)(23,32)(25,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,2,5,8)(3,6,11,9)(4,7,13,16)(10,14,12,15)(17,19,22,27)(18,23,26,32)(20,21,28,30)(24,31,29,25);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,24,26,29)(19,21,27,30)(23,31,32,25);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21)(18,25)(19,28)(20,27)(22,30)(23,24)(26,31)(29,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,30)(27,32)(28,29)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831208);