#416171

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,11)(8,9)(10,12)(13,14,18,23,19,22,17,32)(15,24,27,35,28,39,26,20)(16,29,30,36,31,37,33,21)(25,34,40,43,41,44,42,38);(1,2)(3,4)(6,9)(10,11)(13,15,18,27,19,28,17,26)(14,20,32,39,22,35,23,24)(16,25,33,42,31,41,30,40)(21,34,29,43,36,44,37,38);(2,4)(13,16)(14,21)(15,25)(17,30)(18,33)(19,31)(20,34)(22,36)(23,37)(24,38)(26,40)(27,42)(28,41)(29,32)(35,44)(39,43);(5,7,8,12)(6,10,9,11)(14,22)(20,35)(21,36)(23,32)(24,39)(29,37)(34,44)(38,43)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 323367);