#426559

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi

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, 333755);