#383918

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1838 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)(19,28)(20,25)(21,31)(22,32)(23,27)(24,29)(26,30);(1,2)(3,4)(9,11,12,15)(10,13,14,16)(17,19,22,29,23,21,20,30)(18,24,25,28,27,26,32,31);(2,4)(10,14)(13,16)(17,20,23,22)(18,25,27,32)(19,29,21,30)(24,31,26,28);(10,14)(13,16)(17,21,23,19)(18,26,27,24)(20,30,22,29)(25,28,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291114);