#414680

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1508 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,e^2=i,f^2=i,g^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,g^b=gi,e^c=ei,f^c=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, 321876);