#418206

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325402);