#232146

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=de,f^b=fh,g^b=gh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139342);