#456502

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1168 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,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, 363698);