#116550

order	512
nilpotency class	3
isoclinism	isoclinic class of #98904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
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

678 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:

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

Polycyclic presentation:

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

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, 23746);