#8696144

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	#51948	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,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, 8603340);