#471440

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1149 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)(7,12)(8,21)(9,15)(10,13)(11,16)(14,24)(17,27)(18,32)(19,28)(20,25)(22,29)(23,30)(26,34)(31,35)(33,36);(5,7,10,19,9,17,11,20)(6,12,16,25,15,27,13,28)(8,18,24,34,22,31,23,33)(14,26,21,32,30,36,29,35);(2,4)(6,13)(7,17)(8,22)(10,11)(12,25)(14,21)(15,16)(18,31)(26,32)(27,28)(29,30)(35,36);(5,8)(6,14)(7,18)(9,22)(10,24)(11,23)(12,26)(13,29)(15,30)(16,21)(17,31)(19,34)(20,33)(25,32)(27,36)(28,35)

Polycyclic presentation:

2^9|b^2=egh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,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, 378636);