#197951

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1270 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)(7,10)(11,12)(13,14,18,24,19,22,17,23)(15,20,28,36,26,37,27,35)(16,29,33,21,31,40,32,39)(25,41,43,34,30,44,42,38);(5,6,10,12,8,11,7,9)(13,15,17,28,19,26,18,27)(14,20,24,35,22,37,23,36)(16,30,33,43,31,25,32,42)(21,38,29,41,39,34,40,44);(2,4)(6,11)(9,12)(13,16,19,31)(14,21,22,39)(15,25,26,30)(17,33,18,32)(20,34,37,38)(23,40,24,29)(27,43,28,42)(35,44,36,41);(14,22)(15,26)(21,39)(23,24)(25,30)(27,28)(29,40)(42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 105147);