#413828

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 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)(7,18)(8,23)(9,17)(10,15)(11,16)(12,20)(13,21)(14,26)(19,28)(22,29)(24,36)(25,33)(27,32)(30,34)(31,35);(2,4)(5,7)(6,12,16,29)(8,19)(9,20)(10,22)(11,21)(13,15,18,17)(14,28,33,36)(23,30,32,31)(24,27)(25,34)(26,35);(5,7)(6,13,16,18)(8,24)(9,20)(10,22)(11,21)(12,17,29,15)(14,31,33,30)(19,27)(23,28,32,36)(25,35)(26,34);(5,8,10,26,11,27,9,25)(6,14,17,23,16,33,15,32)(7,19,22,35,21,24,20,34)(12,28,13,30,29,36,18,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231517, #72231526, #72231541, …
maximal normal subgroup of	#72231517, #72231526, #72231541, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321024);