#129497

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

866 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 150 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)(7,9)(8,11)(10,12)(13,14,16,21,18,22,29,36,17,20,28,33,30,34,41,43)(15,23,25,35,26,37,39,44,24,19,27,31,38,32,40,42);(6,9)(7,12)(10,11)(13,15)(14,19)(16,27)(17,24)(18,26)(20,23)(21,35)(22,32)(25,28)(29,40)(30,38)(31,33)(34,37)(36,44)(39,41)(42,43);(2,4)(15,24)(19,23)(25,27)(26,38)(31,35)(32,37)(39,40)(42,44)

Polycyclic presentation:

2^9|a^2=f,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dgi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36693);