#212005

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
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

545 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:

(1,2,7,10)(3,6,9,5)(4,12,14,11)(8,15,16,13)(17,18,21,25,22,19,26,24)(20,23,29,31,28,27,32,30);(1,3,5,2,7,9,6,10)(4,12,13,16,14,11,15,8)(20,28)(23,27)(29,32)(30,31);(1,4)(2,8)(3,11)(5,13)(6,15)(7,14)(9,12)(10,16)(17,19)(18,22)(20,27)(21,24)(23,28)(25,26)(29,30)(31,32);(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=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=dhi,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, 119201);