#108001

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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	#379
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 15197);