#119694

order	512
nilpotency class	3
isoclinism	isoclinic class of #10464
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fh,e^c=ehi

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, 26890);