#8521633

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34,39,42)(35,43,45,38)(36,41,37,40)(44,47,46,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,23,27)(18,24,26,21)(20,28,25,32)(22,29,31,30)(33,35)(34,38)(36,44)(37,46)(39,45)(40,47)(41,48)(42,43);(5,13)(8,12)(10,16)(14,15)(19,27)(21,24)(28,32)(29,30)(33,36,39,37)(34,40,42,41)(35,44,45,46)(38,47,43,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,37,39,36)(34,41,42,40)(35,44,45,46)(38,47,43,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,35)(34,38)(36,46)(37,44)(39,45)(40,48)(41,47)(42,43);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,38)(34,35)(36,47)(37,48)(39,43)(40,44)(41,46)(42,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8428829);