#8521591

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	#36987	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,11)(4,13)(5,7)(6,10)(8,12)(9,15)(14,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(34,38)(36,43)(39,44)(42,47);(1,3,7,12)(2,8,5,11)(4,14,15,6)(9,16,13,10)(17,19,23,27)(18,24,26,21)(20,28,25,32)(22,29,31,30)(35,41)(39,44)(42,47)(46,48);(2,5)(8,11)(9,13)(10,16)(19,27)(21,24)(28,32)(29,30)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,15)(2,9,5,13)(3,6,12,14)(8,10,11,16)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,7,2)(3,11,12,8)(4,13,15,9)(6,16,14,10)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,6)(2,10)(3,4)(5,16)(7,14)(8,9)(11,13)(12,15)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^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, 8428787);