#438336

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345532);