#416214

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

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

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, 323410);