#395988

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

1332 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)(5,6,12,8,11,16,10,15)(7,17,26,18,21,28,20,13)(9,22,23,14,25,30,24,29)(19,32,33,31,35,27,34,36);(5,7,10,20,11,21,12,26)(6,13,8,17,16,18,15,28)(9,19,23,33,25,35,24,34)(14,27,22,31,29,32,30,36);(2,4)(5,8,10,16,11,15,12,6)(7,18,20,28,21,13,26,17)(9,22,23,29,25,30,24,14)(19,32,33,36,35,27,34,31);(5,9)(6,14)(7,19)(8,22)(10,23)(11,25)(12,24)(13,27)(15,30)(16,29)(17,31)(18,32)(20,33)(21,35)(26,34)(28,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 303184);