#225224

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 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)(3,5)(4,6)(7,8)(10,14)(11,12)(15,16)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,16)(13,15)(19,25)(22,28)(26,31)(29,32);(2,6)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(9,11)(12,13)(14,16)(18,23)(19,25)(24,30)(26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,c^b=ci,d^b=de,e^b=eh,d^c=dhi,e^d=eh

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, 132420);