#8898946

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284850, #72284967, #72285392
maximal normal subgroup of	#72284850, #72284967, #72285392

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806142);