#392410

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 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)(3,5)(4,6)(7,8)(9,10,15,19,14,17,13,20)(11,21,25,31,22,32,24,16)(12,26,27,33,28,34,29,18)(23,30,37,39,36,40,38,35);(1,3,4,7)(2,5,6,8)(9,11,15,25,14,22,13,24)(10,16,20,32,17,31,19,21)(12,23,29,38,28,36,27,37)(18,30,26,39,33,40,34,35);(2,6)(5,8)(10,17)(11,22)(18,33)(19,20)(23,36)(24,25)(26,34)(37,38);(9,12,13,27,14,28,15,29)(10,18,19,34,17,33,20,26)(11,23,24,37,22,36,25,38)(16,30,21,35,31,40,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299606);