#183349

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,c^b=ci,e^b=eh,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90545);