#400508

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1040 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:

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

Polycyclic presentation:

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

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, 307704);