#8720528

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627724);