#411289

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

934 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318485);