#126944

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 714 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) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,d^b=di,e^b=egi

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, 34140);