#424292

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1515 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)(5,6,10,15,11,16,23,17)(7,14,9,27,19,34,20,12)(8,21,25,33,26,13,29,32)(18,35,24,28,22,31,36,30);(5,7,11,19)(6,12,16,27)(8,22,26,18)(9,10,20,23)(13,30,21,28)(14,15,34,17)(24,29,36,25)(31,33,35,32);(1,2,3,4)(5,8,23,29,11,26,10,25)(6,13,17,33,16,21,15,32)(7,18,20,36,19,22,9,24)(12,28,34,35,27,30,14,31);(5,9,10,19,11,20,23,7)(6,14,15,27,16,34,17,12)(8,24,25,22,26,36,29,18)(13,31,32,30,21,35,33,28)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,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, 331488);