#181737

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×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	C₂²×#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1353 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,22,26,23,19,21,25)(20,24,28,31,30,27,29,32);(1,3,5,14,6,12,4,13)(2,7,8,16,10,15,9,11)(17,19,23,18)(20,24,30,27)(21,26,22,25)(28,31,29,32);(3,12)(7,15)(11,16)(13,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4,6,5)(2,8,10,9)(3,13,12,14)(7,16,15,11)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,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, 88933);