#388437

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
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×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ehi,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, 295633);