#109992

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,8)(4,6)(5,7)(9,10,14,17,15,11,13,19)(12,20,21,16,22,24,23,18);(1,3,4,2,5,6,8,7)(9,11,15,10)(12,18,22,16)(13,17,14,19)(20,23,24,21);(9,12,14,21,15,22,13,23)(10,16,17,24,11,18,19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209999
maximal normal subgroup of	#72209999
maximal quotient of	#72239955, #72239960, #72240023, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17188);