#212003

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

553 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119199);