#499078

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1220 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)(7,10)(8,22)(9,15)(11,16)(12,18)(13,17)(14,25)(19,26)(20,27)(21,29)(23,28)(24,34)(30,35)(31,33)(32,36);(5,7)(6,13)(8,23)(9,20)(10,15)(11,17)(12,21)(14,24)(16,29)(18,27)(19,25)(22,31)(26,32)(28,34)(30,33)(35,36);(2,4)(5,8,9,24,12,26,11,30)(6,14,16,23,18,33,15,32)(7,19,17,34,21,22,20,35)(10,28,27,25,29,36,13,31);(5,9,12,11)(6,15,18,16)(7,20,21,17)(8,24,26,30)(10,27,29,13)(14,32,33,23)(19,35,22,34)(25,36,31,28)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gh,h^b=hi,f^c=fhi,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307089
maximal normal subgroup of	#72307089

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406274);