#102319

order	512
nilpotency class	3
isoclinism	isoclinic class of #102164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

554 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=e,d^2=gh,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,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, 9515);