#116899

order	512
nilpotency class	3
isoclinism	isoclinic class of #116897
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

633 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 633 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18)(19,25,27,32)(20,23)(21,24)(22,26,30,29)(28,31);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(20,28)(22,30)(23,31)(26,29);(17,19)(18,22)(20,29)(21,27)(23,32)(24,30)(25,31)(26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24095);