#403521

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1252 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)(6,7)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,30,32,31);(5,6,8,7)(9,10,12,14)(11,15,16,13)(17,19)(18,22)(20,27)(21,28)(23,31)(24,26)(25,30)(29,32);(9,11,12,16)(10,13,14,15)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(2,4)(5,7,8,6)(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=gi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,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, 310717);