#109394

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

574 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,f^b=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, 16590);