#98160

order	512
nilpotency class	3
isoclinism	isoclinic class of #98134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#36504	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

886 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239731, #72239736, #72239741
maximal normal subgroup of	#72239731, #72239736, #72239741

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5356);