#305761

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

680 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=fi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212957);