#413883

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1492 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,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, 321079);