#198498

order	512
nilpotency class	3
isoclinism	isoclinic class of #198498
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

1295 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105694);