#109842

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240066, #72240068
maximal normal subgroup of	#72240066, #72240068
maximal quotient of	#72240032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17038);