#472283

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

951 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ef,e^2=i,f^2=i,g^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,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi,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, 379479);