#9910988

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	#37682	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4562 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(18,19)(22,35)(24,27)(25,28)(31,45)(33,36)(37,41)(43,46);(2,5)(6,7)(10,13)(14,15)(17,18,23,19)(20,24,32,27)(21,25,34,28)(22,26,35,29)(30,37,44,41)(31,38,45,42)(33,39,36,40)(43,47,46,48);(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,24,32,27)(21,28,34,25)(22,29,35,26)(30,37,44,41)(31,38,45,42)(33,40,36,39)(43,47,46,48);(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,30)(22,31)(23,32)(25,37)(26,38)(28,41)(29,42)(33,43)(34,44)(35,45)(36,46)(39,47)(40,48);(1,2)(3,7)(4,5)(6,8)(17,21,23,34)(18,25,19,28)(20,30,32,44)(22,36,35,33)(24,37,27,41)(26,40,29,39)(31,46,45,43)(38,48,42,47);(1,3)(2,6)(4,8)(5,7)(17,22)(18,26)(19,29)(20,31)(21,33)(23,35)(24,38)(25,39)(27,42)(28,40)(30,43)(32,45)(34,36)(37,47)(41,48)(44,46)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=h,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, 9818184);