#305780

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

570 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)(10,13)(14,16)(17,18,22,24)(19,27,29,25)(20,23,30,32)(21,31,28,26);(10,14)(12,15)(13,16)(17,19,28,30,22,29,21,20)(18,23,31,25,24,32,26,27);(1,3,4,8)(2,5,6,7)(9,10,11,13)(12,16,15,14)(18,24)(23,32)(25,27)(26,31);(17,20,21,29,22,30,28,19)(18,25,26,23,24,27,31,32)

Polycyclic presentation:

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

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, 212976);