#108393

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#376
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

654 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)(5,6,8,10,7,9,11,12)(13,14)(15,24,27,38)(16,21)(17,22)(18,32)(19,23)(20,26,34,28)(25,37,40,44)(29,35)(30,42)(31,36)(33,39,43,41);(6,9)(10,12)(13,15,18,28,19,27,17,26)(14,20,22,24,23,34,32,38)(16,25,30,41,31,40,29,39)(21,33,35,37,36,43,42,44);(2,4)(13,16,17,29,19,31,18,30)(14,21,22,35,23,36,32,42)(15,25,26,39,27,40,28,41)(20,33,24,37,34,43,38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=di,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,g^a=gh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15589);