#8728166

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	#59685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8635362);