#411517

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

940 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)(5,6,11,14,9,15,10,16)(7,12,24,27,18,28,19,26)(8,20,23,31,21,30,22,13)(17,32,35,29,33,36,34,25);(1,2,4,3)(5,7,10,24,9,18,11,19)(6,12,16,27,15,28,14,26)(8,17,22,35,21,33,23,34)(13,29,30,36,31,25,20,32);(1,3,4,2)(5,8,9,21)(6,13)(7,17,18,33)(10,22,11,23)(12,25)(14,20)(15,31)(16,30)(19,34,24,35)(26,36)(27,32)(28,29);(6,14,15,16)(8,21)(12,26,28,27)(13,30,31,20)(17,33)(22,23)(25,32,29,36)(34,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=g,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, 318713);