#146772

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1054 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)(7,16)(8,12)(9,25)(10,14)(11,17)(13,22)(15,23)(18,28)(19,32)(20,27)(21,29)(24,30)(26,34)(31,35)(33,36);(5,7)(6,11,14,28)(8,21)(9,26)(10,20)(12,27,30,16)(13,32,31,34)(15,33,25,29)(17,35)(18,22)(19,23)(24,36);(6,12)(8,22)(9,23)(11,27)(13,15)(14,30)(16,28)(17,36)(18,21)(19,26)(24,35)(25,31)(29,34)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231475, #72231499, #72231513, …
maximal normal subgroup of	#72231475, #72231499, #72231513, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53968);