#414762

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
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

1358 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 321958);