#199484

order	512
nilpotency class	3
isoclinism	isoclinic class of #199221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3109	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239729, #72239737, #72239757
maximal normal subgroup of	#72239729, #72239737, #72239757
maximal quotient of	#72257822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106680);