#479250

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

671 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)(7,16)(8,13,23,32)(9,11,24,14)(12,26)(17,28)(18,34)(19,29)(20,36)(21,30,22,25)(27,35)(31,33);(6,11)(7,17)(9,24)(12,16)(13,30)(14,15)(18,33)(21,22)(25,32)(27,34)(28,29)(31,36);(5,7,9,19,10,17,24,26)(6,12,14,28,15,29,11,16)(8,20,21,33,23,35,22,18)(13,31,25,36,32,34,30,27);(2,4)(5,8,9,22,10,23,24,21)(6,13,11,25,15,32,14,30)(7,18,19,35,17,33,26,20)(12,27,16,31,29,36,28,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386446);