#6434816

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342012);