#455217

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
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×#472	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1305 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)(5,6,11,14,10,15,9,23)(7,16,24,27,17,29,19,12)(8,20,25,28,22,35,21,13)(18,32,33,36,31,26,34,30);(5,7,9,19,10,17,11,24)(6,12,14,16,15,27,23,29)(8,18,21,34,22,31,25,33)(13,26,20,30,28,32,35,36);(7,17)(8,21)(9,11)(13,20)(14,23)(16,29)(18,33)(22,25)(26,30)(28,35)(31,34)(32,36);(2,4)(5,8,11,25,10,22,9,21)(6,13,23,35,15,28,14,20)(7,18,24,33,17,31,19,34)(12,26,29,36,27,32,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362413);