#412844

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=e,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,c^b=cg,g^b=gi,g^c=gi

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, 320040);