#108557

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

494 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27,29,23)(20,31)(21,26)(22,25)(24,32)(28,35,41,39)(30,40,43,36)(33,44)(34,38)(37,45)(42,46,48,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,30,22,29,34,43)(18,23,26,36,25,27,38,40)(20,28,33,42,32,41,45,48)(24,35,37,46,31,39,44,47);(3,9)(6,12)(10,15)(13,16)(17,20)(18,24,25,31)(19,28)(21,33)(22,32)(23,35,27,39)(26,37,38,44)(29,41)(30,42)(34,45)(36,46,40,47)(43,48)

Polycyclic presentation:

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

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, 15753);