#183015

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

913 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)(3,11,12,16)(4,10,6,8)(7,14,15,13)(17,18,21,26,22,27,30,25)(19,20,24,32,28,31,29,23);(1,3,6,14,5,12,4,13)(2,7,10,16,9,15,8,11)(17,19,21,24,22,28,30,29)(18,23,26,20,27,32,25,31);(17,20,30,23,22,31,21,32)(18,24,25,19,27,29,26,28);(3,12)(7,15)(11,16)(13,14)(19,28)(20,31)(23,32)(24,29)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=ghi,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90211);