#147871

order	512
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(5,6,7,8)(10,15)(11,18)(12,22)(13,21)(16,20)(17,24)(19,23);(1,2)(3,4)(9,10)(11,19)(12,20)(13,23)(14,17)(15,18)(16,21)(22,24);(2,4)(9,10)(11,20)(12,19)(13,23)(14,17)(15,22)(16,21)(18,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213167, #72213261, #72213525
maximal normal subgroup of	#72213167, #72213261, #72213525
maximal quotient of	#72213148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55067);