#401940

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309136);