#379858

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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

1652 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)(10,13)(14,16)(17,18)(19,27)(20,23)(21,25)(22,31)(24,29)(26,30)(28,32);(1,2)(3,4)(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,23,25,27)(22,28,30,29)(24,26,32,31);(2,4)(9,11)(10,14)(12,15)(13,16)(17,20,21,19)(18,24,25,32)(22,28,30,29)(23,26,27,31);(11,15)(14,16)(19,20)(23,27)(24,32)(28,29)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=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, 287054);