#9054421

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	#52904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8961617);