#455117

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1502 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)(5,6,10,15,11,16,9,13)(7,17,21,28,18,26,20,12)(8,22,24,14,25,29,23,30)(19,27,33,31,32,36,34,35);(5,7,9,20,11,18,10,21)(6,12,15,17,16,28,13,26)(8,19,23,34,25,32,24,33)(14,27,29,31,30,36,22,35);(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(17,28)(19,33)(22,29)(24,25)(31,35)(32,34);(2,4)(5,8,10,24,11,25,9,23)(6,14,13,22,16,30,15,29)(7,19,21,33,18,32,20,34)(12,27,26,35,28,36,17,31)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,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, 362313);