#220669

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	#61409	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

871 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 127865);