#380706

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,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, 287902);