#220711

order	512
nilpotency class	3
isoclinism	isoclinic class of #220564
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	#61217	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127907);