#8751666

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8658862);