#134275

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,e^c=ei,f^c=fgi

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, 41471);