#203070

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322077, #72322165, #72322213
maximal normal subgroup of	#72322077, #72322165, #72322213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110266);