#126686

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33882);