#405921

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1174 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)(5,6,11,17)(7,18)(8,24)(9,26)(10,15,29,16)(12,22)(13,14)(19,35,36,30)(20,25,33,34)(21,32)(23,31)(27,28);(1,2,4,3)(5,7,10,21,11,23,29,22)(6,12,16,31,17,32,15,18)(8,25,26,36,27,34,14,19)(9,20,24,30,13,33,28,35);(1,3,4,2)(5,8,10,14,11,27,29,26)(6,13,15,28,17,9,16,24)(7,19,21,34,23,36,22,25)(12,30,18,33,32,35,31,20);(5,9)(6,14)(7,20)(8,16)(10,24)(11,13)(12,19)(15,27)(17,26)(18,34)(21,30)(22,35)(23,33)(25,31)(28,29)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313117);