#305415

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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

696 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)(10,14)(15,16)(17,18)(19,26)(20,28)(21,30)(22,25)(23,27)(24,31)(29,32);(10,15)(11,13)(14,16)(17,19,22,20)(18,23,25,32)(21,29,31,27)(24,28,30,26);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,15,13,16);(9,11,12,13)(10,15,14,16)(17,20,22,19)(18,23,25,32)(21,27,31,29)(24,28,30,26)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212611);