#380778

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,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, 287974);