#411183

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	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)(5,6,11,14,10,15,9,16)(7,17,26,28,20,12,19,29)(8,21,25,31,23,30,24,13)(18,33,36,27,22,35,34,32);(1,2,4,3)(5,7,9,26,10,20,11,19)(6,12,16,29,15,17,14,28)(8,22,24,34,23,18,25,36)(13,27,30,35,31,32,21,33);(5,8,10,23)(6,13)(7,18,20,22)(9,24,11,25)(12,27)(14,21)(15,31)(16,30)(17,32)(19,34,26,36)(28,33)(29,35);(1,3,4,2)(6,14,15,16)(8,23)(12,28,17,29)(13,30,31,21)(18,22)(24,25)(27,35,32,33)(34,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=g,d^2=e,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,f^b=fi,d^c=di

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, 318379);