#9160816

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(2,7)(4,10)(9,13)(12,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,7)(3,8)(9,13)(11,16)(18,22)(23,25)(24,28)(31,32)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068012);