#220694

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
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	#61329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127890);