#205593

order	512
nilpotency class	3
isoclinism	isoclinic class of #205546
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305430, #72305433, #72305436, …
maximal normal subgroup of	#72305430, #72305433, #72305436, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112789);