#376273

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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)(5,6,12,17,10,16,11,33)(7,18,23,26,21,13,22,34)(8,24,28,14,27,35,29,20)(9,25,32,36,30,15,31,19);(1,2,4,3)(5,7,10,21)(6,13,16,18)(8,25,27,15)(9,20,30,14)(11,22,12,23)(17,34,33,26)(19,28,36,29)(24,31,35,32);(1,2,4,3)(5,8,10,27)(6,14,16,20)(7,19,21,36)(9,26,30,34)(11,29,12,28)(13,31,18,32)(15,23,25,22)(17,35,33,24);(1,3,4,2)(5,9,11,31,10,30,12,32)(6,15,33,36,16,25,17,19)(7,20,22,35,21,14,23,24)(8,26,29,18,27,34,28,13)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283469);