#146802

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1762
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240727, #72240728, #72240733, …
maximal normal subgroup of	#72240727, #72240728, #72240733, …
maximal quotient of	#72308704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53998);