#110204

order	512
nilpotency class	3
isoclinism	isoclinic class of #110191
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224692, #72224702, #72224704, …
maximal normal subgroup of	#72224692, #72224702, #72224704, …
maximal quotient of	#72257737, #72257745

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17400);