#181219

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235666, #72238986, #72238987, …
maximal normal subgroup of	#72235666, #72238986, #72238987, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88415);