#183518

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
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	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

689 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90714);