#8728156

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,24)(21,29)(22,26)(23,31)(27,30)(28,32)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,27)(22,29)(23,30)(24,32)(25,26)(28,31)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(3,10)(6,7)(11,16)(14,15)(17,20,22,31)(18,23,26,24)(19,27,29,28)(21,30,25,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,24)(19,28)(21,30)(22,31)(23,26)(25,32)(27,29)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,28)(26,29)(31,32)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,25)(20,27)(21,26)(22,29)(23,32)(24,30)(28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,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, 8635352);