#225731

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,d^c=di,e^d=eh

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, 132927);