#220670

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61410	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127866);