#380215

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, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi,h^c=hi

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, 287411);