#410912

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
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

844 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)(3,6)(4,5)(7,8)(17,18,22,24)(19,21,26,31)(20,27,30,25)(23,28,32,29);(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,19,21,24,22,26,31,18)(20,25,28,32,30,27,29,23);(9,11,12,16)(10,13,14,15)(17,20)(18,23)(19,25)(21,28)(22,30)(24,32)(26,27)(29,31);(1,3,4,7)(2,6,5,8)(17,19,21,24,22,26,31,18)(20,25,28,32,30,27,29,23)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,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, 318108);