#211968

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
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

541 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh,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, 119164);