#409170

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

770 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)(5,6,9,14,11,15,10,24)(7,16,17,27,19,25,20,12)(8,13,22,29,21,28,23,34)(18,30,31,36,33,35,32,26);(5,7,9,20,11,19,10,17)(6,12,14,25,15,27,24,16)(8,18,22,32,21,33,23,31)(13,26,29,35,28,36,34,30);(1,2,4,3)(7,17,19,20)(8,21)(12,25,27,16)(13,28)(18,32,33,31)(22,23)(26,30,36,35)(29,34);(1,3,4,2)(5,8,9,22,11,21,10,23)(6,13,14,29,15,28,24,34)(7,18,20,32,19,33,17,31)(12,26,25,35,27,36,16,30)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,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, 316366);