#396328

order	512
nilpotency class	3
isoclinism	isoclinic class of #394034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1426 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=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, 303524);