#109996

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	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

533 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,f^2=hi,g^2=i,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	C₂×#109996, #72240066, #72240067, …
maximal normal subgroup of	C₂×#109996, #72240066, #72240067, …
maximal quotient of	C₂×#109996, #72240054, #72240055, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17192);