#214003

order	512
nilpotency class	3
isoclinism	isoclinic class of #214000
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121199);