#198769

order	512
nilpotency class	3
isoclinism	isoclinic class of #198528
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	C₂³×C₄	C₂⁴
prime radical	#60734	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 105965);