#8521637

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	#36808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,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, 8428833);