#494931

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

830 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250241, #72250266, #72250324, …
maximal normal subgroup of	#72250241, #72250266, #72250324, …
maximal quotient of	#72319254, #72319900, C₂×#494931

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402127);