#220703

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	#61410	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=hi,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, 127899);