#441227

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1231 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)(7,11,20,30)(8,13,24,33)(9,16)(10,14)(12,25,32,26)(15,17)(18,35)(19,27)(21,29,22,34)(23,28)(31,36);(5,7,17,21,9,20,10,22)(6,11,15,29,16,30,14,34)(8,23,25,18,24,19,26,36)(12,31,13,28,32,35,33,27);(2,4)(5,8)(6,12,16,32)(7,18)(9,24)(10,25)(11,27,30,28)(13,15,33,14)(17,26)(19,22)(20,36)(21,23)(29,35,34,31);(5,8)(6,13,16,33)(7,19)(9,24)(10,26)(11,28,30,27)(12,15,32,14)(17,25)(18,22)(20,23)(21,36)(29,35,34,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348423);