#210320

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117516);