#396384

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 303580);