#108500

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
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	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

646 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)(5,6,7,9,8,10,11,12)(13,14)(15,25,28,20)(16,30,32,38)(17,23)(18,24)(19,22)(21,31,37,33)(26,40)(27,36,29,35)(34,41)(39,42)(43,44);(13,15,18,29,19,28,17,27)(14,20,24,36,22,25,23,35)(16,26,33,42,32,43,31,41)(21,34,38,40,37,39,30,44);(2,4)(13,16)(14,21,22,37)(15,26)(17,31)(18,33)(19,32)(20,34,25,39)(23,30,24,38)(27,41)(28,43)(29,42)(35,44,36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,f^c=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, 15696);