#413130

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=hi,e^2=i,f^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,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320326);