#8521609

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	#52330	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=g,f^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, 8428805);