#376265

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283461);