#413502

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320698);