#397202

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1439 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:

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

Polycyclic presentation:

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

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, 304398);