#202085

order	512
nilpotency class	3
isoclinism	isoclinic class of #201928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1165 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109281);