#8926780

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	#35676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5234 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833976);