#459405

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,h^d=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, 366601);