#197966

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,18,24,19,22,17,23)(15,20,28,36,26,34,27,35)(16,29,32,21,30,38,31,37)(25,39,41,33,42,43,40,44);(5,6,7,9,8,10,11,12)(13,15,18,27,19,26,17,28)(14,20,23,36,22,34,24,35)(16,25,31,41,30,42,32,40)(21,33,38,39,37,44,29,43);(2,4)(13,16,19,30)(14,21,22,37)(15,25,26,42)(17,32,18,31)(20,33,34,44)(23,38,24,29)(27,41,28,40)(35,43,36,39);(6,10)(9,12)(14,22)(20,34)(21,37)(23,24)(29,38)(33,44)(35,36)(39,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 105162);