#481952

order	512
nilpotency class	4
isoclinism	isoclinic class of #11796
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389148);