#182089

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1015 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16);(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,b^a=be,g^a=gh,e^b=eh,d^c=di

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268383, #72268390, #72268407, …
maximal normal subgroup of	#72268383, #72268390, #72268407, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89285);