#106839

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

894 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)(5,6,10,11,9,7,8,12)(13,14)(15,23,25,19)(16,27,28,20)(17,22)(18,21)(24,35)(26,36,39,32)(29,40,41,33)(30,34)(31,37)(38,43)(42,44);(2,4)(5,7,9,6)(8,12,10,11)(13,15,17,26,18,25,30,39)(14,19,22,32,21,23,34,36)(16,24,29,38,28,37,41,44)(20,31,33,42,27,35,40,43);(5,8,9,10)(6,11,7,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,31)(21,27)(22,33)(23,35)(25,37)(26,38)(30,41)(32,42)(34,40)(36,43)(39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=eh,f^b=fhi,d^c=dh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14035);