#102180

order	512
nilpotency class	3
isoclinism	isoclinic class of #102147
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

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

normal subgroups only

This group has 382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9376);