#455116

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
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 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)(8,22,24,30,25,29,23,14)(19,33,34,36,32,27,35,31);(5,7,9,20,11,18,10,21)(6,12,15,17,16,28,13,26)(8,19,23,35,25,32,24,34)(14,27,22,31,30,33,29,36);(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(17,28)(19,34)(22,29)(24,25)(31,36)(32,35);(2,4)(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,13,29)(7,19,20,35,18,32,21,34)(12,27,17,31,28,33,26,36)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,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, 362312);