#120669

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#397
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

560 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)(6,11)(7,12)(8,9)(13,14)(15,24)(16,29)(17,33)(18,22)(19,23)(20,26)(21,31)(25,37)(27,36)(28,41)(30,40)(32,38)(34,39)(35,43)(42,44);(5,6,9,7,10,12,8,11)(13,15,17,27,19,26,18,28)(14,20,22,36,23,24,33,41)(16,30,31,42,32,43,34,25)(21,37,29,35,39,44,38,40);(5,7,8,6,10,11,9,12)(13,16)(14,21)(15,25,26,42)(17,34)(18,31)(19,32)(20,35,24,40)(22,38)(23,39)(27,43,28,30)(29,33)(36,37,41,44)

Polycyclic presentation:

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

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, 27865);