#198952

order	512
nilpotency class	3
isoclinism	isoclinic class of #198789
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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238931, #72238934, #72238944
maximal normal subgroup of	#72238931, #72238934, #72238944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106148);