#408530

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fh,c^2=ei,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=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, 315726);