#108912

order	512
nilpotency class	3
isoclinism	isoclinic class of #108898
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 487 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18,21,27,23,28,22,26)(19,29,32,40,33,39,31,24)(20,34,35,25,37,41,36,42)(30,38,45,43,46,48,44,47);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19,22,31,23,33,21,32)(18,24,26,39,28,40,27,29)(20,30,36,44,37,46,35,45)(25,38,34,47,42,48,41,43);(17,20,21,35,23,37,22,36)(18,25,26,34,28,42,27,41)(19,30,32,45,33,46,31,44)(24,38,39,47,40,48,29,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16108);