#146763

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

984 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)(7,18)(8,12)(9,26)(10,28)(11,16)(13,19)(14,25)(15,22)(17,24)(20,33)(21,30)(23,32)(27,35)(29,34)(31,36);(6,12,16,26)(8,22)(9,25)(10,24)(13,30,31,18)(14,28,15,17)(19,23)(20,29)(27,36)(32,34,35,33);(5,7)(6,13)(8,23)(9,27)(10,29)(11,21)(12,30)(14,33)(15,34)(16,31)(17,35)(18,26)(19,22)(20,24)(25,36)(28,32)

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=ehi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231513, #72231524, #72231547, …
maximal normal subgroup of	#72231513, #72231524, #72231547, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53959);