#408571

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
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

816 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315767);