#8069691

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	#48269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976887);