#383411

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1440 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

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, 290607);