#471459

order	512
nilpotency class	3
isoclinism	isoclinic class of #456884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 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)(5,6,11,9)(7,15,20,30)(8,13,24,25)(10,26,28,14)(12,19,17,18)(16,31,35,33)(21,29,34,36)(22,23,32,27);(5,7,11,20)(6,12,9,17)(8,21)(10,19,28,18)(13,31)(14,30,26,15)(16,27)(22,36)(23,35)(24,34)(25,33)(29,32);(2,4)(5,8)(6,13)(7,16)(9,25)(10,27)(11,24)(12,29)(14,32)(15,31)(17,36)(18,21)(19,34)(20,35)(22,26)(23,28)(30,33);(5,9,11,6)(7,17,20,12)(8,22,24,32)(10,14,28,26)(13,27,25,23)(15,19,30,18)(16,33,35,31)(21,36,34,29)

Polycyclic presentation:

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

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, 378655);