#8751677

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 8658873);