#380322

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

1308 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)(11,14)(12,13)(15,16)(18,22)(19,25)(21,27)(23,30)(26,32)(28,31);(1,2)(3,4)(10,13)(14,16)(17,18,20,22)(19,23,26,28)(21,24,27,29)(25,31,32,30);(2,4)(11,15)(14,16)(17,19,21,25,20,26,27,32)(18,23,24,31,22,28,29,30);(9,11,12,15)(10,14,13,16)(18,22)(19,26)(24,29)(25,32)

Polycyclic presentation:

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

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, 287518);