#102184

order	512
nilpotency class	3
isoclinism	isoclinic class of #102154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

354 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9380);