#198219

order	512
nilpotency class	3
isoclinism	isoclinic class of #198144
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1077 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105415);