#106124

order	512
nilpotency class	3
isoclinism	isoclinic class of #8608
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,22,23,16)(13,18)(14,17)(19,31,32,26)(20,28)(21,27)(24,35,36,29)(25,30)(33,39,40,37)(34,38);(2,5,7,6)(9,11,13,20,14,21,25,34)(10,15,18,28,17,27,30,38)(12,19,24,33,23,32,36,40)(16,26,29,37,22,31,35,39);(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,22)(18,29)(20,33)(21,32)(25,36)(27,31)(28,37)(30,35)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318993, #72343136, #72343141
maximal normal subgroup of	#72318993, #72343136, #72343141
maximal quotient of	#72319258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13320);